WORM  GEARING 


McGraw-Hill  BookGompany 


Electrical  World         Tte  Engineering  andMining  Journal 
Engineering  Record  Engineering  News 

Railway  Age  Gazette  American  Machinist 

Signal  LnginGer  American  Engineer 

Electric  Railway  Journal  Coal  Age 

Metallurgical  and  Chemical  Engineering  Power 


Frontspiece. 
A  Fifteenth  Century  Worm  Gear  from  an  Etching  by  Albrecht  Diirer. 


WORM  GEARING 


BY 
HUGH  KERR  THOMAS 

A.    M.   I.   MECH.    E. 


McGRAW-HILL    BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1913 


COPYRIGHT,  1913,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY 


THE. MAPLE. PRESS- YORK. PA 


PREFACE 

In  the  following  work  an  attempt  has  been  made,  it  is 
believed  for  the  first  time,  to  deal  exhaustively  with  a  little 
understood  branch  of  applied  mechanics.  A  complete  analysis 
of  the  principles  of  the  design  of  worm  gearing  has  been  made, 
and,  primarily,  this  has  been  treated  in  its  application  to  the 
rear  axles  of  automobiles.  For  a  number  of  years  this  type 
of  gearing  has  enjoyed  considerable  popularity  in  Great  Britain, 
and  its  wider  use  is  daily  becoming  more  general. 

From  his  experience  with  his  own  staff,  as  well  as  with  many 
professional  acquaintances,  the  author  believes  that  the  book 
will  give  information,  which,  although  possibly  possessed  by  a 
few  engineers,  has  not  hitherto  been  accessible  to  the  designer 
and  draughtsman  in  anything  like  a  complete  form. 

An  important  use  for  worm  gearing  is  rapidly  developing 
in  connection  with  the  reduction  gear  of  steam  turbines  for 
marine  propulsion;  while  these  are,  mechanically  speaking, 
simpler  problems,  by  reason  of  their  greater  stability,  than  the 
gears  of  an  automobile,  precisely  the  same  rules  can  be  applied, 
and  the  same  thing  may  be  said  of  gears  for  driving  line  shaft- 
ing from  electric  motors. 

A  brief  outline  of  the  work  was  given  in  two  papers  contrib- 
uted by  the  author  to  the  "Automobile  Engineer"  for  May  and 
June,  1912,  and  some  of  the  formulae  there  published,  with 
much  amplification,  have  been  incorporated  in  the  text.  The 
subject  is  a  complicated  one,  demanding  lengthy  explanations, 
but  by  confining  the  calculations  to  the  use  of  elementary  alge- 
bra and  trigonometry,  it  is  hoped  that  the  solution  of  the  vari- 
ous problems  has  been  as  far  as  possible  simplified. 

The  literature  on  the  subject  is  extremely  meagre;  where 
it  has  been  consulted,  reference  to  the  author  has  been  given  in 
the  text;  with  very  few  exceptions,  however,  the  work  is  entirely 
original,  and  the  rules  given  have  been  in  every  case  referred  to 
practical  experiments  for  verification. 

vii 


257955 


viii  PREFACE 

The  author  wishes  to  acknowledge  many  valuable  sugges- 
tions in  the  progress  of  the  work  from  his  assistants,  John 
Younger,  B.  Sc.,  Lewis  P.  Kalb,  M.  E.,  and  C.  P.  Schwarz, 
D.  Sc.,  to  the  former  of  whom  is  due  the  method  for  determin- 
ing the  width  of  the  worm  wheel. 

He  is  also  indebited  to  A.  L.  Cox  for  valuable  assistance  in 
reading  the  proofs  and  preparing  the  index. 

H.  KERR  THOMAS. 

LONDON,  January,  1913. 


CONTENTS 


PAGE 
PREFACE v 

CHAPTER  I 
INTRODUCTORY     1 

CHAPTER  II 
CHOICE  OF  MATERIALS  AND  METHODS  OF  MANUFACTURE 5 

CHAPTER  III 
DEFINITIONS  AND  SYMBOLS 8 

CHAPTER  IV 
PRELIMINARY  PROPORTIONS 13 

CHAPTER  V 
PRESSURE  ANGLE  AND  FORM  OF  THREAD 22 

CHAPTER  VI 

STRENGTH  OF  WORM  WHEEL  TEETH      34 

CHAPTER  VII 

STRESSES  IN  WORM  GEARING 37 

CHAPTER  VIII 
THE  WIDTH  OF  THE  WORM-WHEEL 47 

CHAPTER  IX 
THE  TEMPERATURE  COEFFICIENT 58 

CHAPTER  X 
EFFICIENCY  OF  WORM  GEARING 63 

CHAPTER  XI 

GENERAL  POINTS  OF  DESIGN  OF  MOUNTING 76 

CHAPTER  XII 

RECAPITULATION  OF  FORMULA  USED 81 

INDEX  .  85 


IX 


WORM  GEARS 

CHAPTER  I 

INTRODUCTORY. 

Worm,  or  more  properly  screw,  gearing  is  of  great  antiquity; 
the  word  screw,  directly  derived  from  the  Danish  scrue,  is  defined 
in  the  Encyclopaedia  Britannica  (llth  edition)  as  "a  cylindrical 
or  conical  piece  of  wood  or  metal  having  a  groove  running 
spirally  round  it."  Such  a  spiral  was  first  studied  geometrically 
by  Archimedes  (287-212  B.  c.)  and  described  in  his  work  Uepl 
cXiKWj  which  deals  in  28  propositions  with  various  mathe- 
matical problems  arising  out  of  the  construction  of  a  helix.  Un- 
fortunately but  too  many  inventions  of  the  early  engineers  have 
passed  out  of  human  knowledge,  and  the  earliest  form  of  worm 
gearing  in  which  an  Archimedian  spiral  was  employed  to  rotate 
a  toothed  wheel  is  not  now  on  record.  We  have,  however,  a 
series  of  curious  drawings  by  the  German  artist,  Albrecht  Diirer 
(1471-1528),  which  were  engraved  on  wood  by  the  Master,  to 
the  order  of  the  Emperor  Maximillian.  It  is  on  record  that  His 
Majesty,  conceiving  for  his  own  aggrandizement  a  Triumphal 
Procession,  commissioned  his  favorite  painter  to  prepare 
designs  of  the  emblematic  cars  which  were  intended  to  figure 
therein.  A  number  of  these  designs  are  still  in  existence  and 
one  is  reproduced  here  in  the  frontispiece.  It  will  be  seen  that 
the  vehicle,  which  is  of  heroic  proportions,  is  propelled  by  all 
four  wheels,  each  actuated  by  a  perfect  worm  gear.  It  is  thus 
a  "four  wheel  driven"  car  in  the  most  modern  sense  of  the 
word  and  is  at  the  same  time  undoubtedly  the  earliest  worm 
driven  vehicle  of  which  any  record  exists.  Unfortunately,  we 
can  hardly  believe  that  so  ponderous  a  machine  could  have 
ever  been  propelled  by  means  of  such  obviously  inefficient 
gearing  operated  by  only  "four  man  power."  It  is  not  known 

1 


2  WORM  GEARS 

whether  this  remarkable  car  was  ever  made,  but  it  is  certain 
from  the  existing  prints  that  Diirer  must  have  seen,  somewhere 
or  other,  a  worm  and  wheel  from  which  his  elaborate  design 
was  copied. 

From  the  fifteenth  to  the  nineteenth  century,  almost  all 
gearing  was  made  of  wood,  and  in  old  treatises  on  gearing, 
numerous  examples  of  wooden  worm  gears  are  illustrated. 
Several  examples  of  worm-driven  traction  engines  are  to  be 
found  in  the  records  of  the  British  Patent  Office,  and  more  than 
one  are  for  the  propulsion  of  tram  cars,  but  as  we  know  it  to-day, 
worm  gear,  which  had  been  used  for  many  years  in  various 
forms,  was  first  applied  to  the  purpose  of  driving  the  road 
wheels  of  automobiles  by  Mr.  F.  W.  Lanchester,  who,  employ- 
ing it  in  his  first  cars  designed  and  built  before  the  close  of  the 
nineteenth  century,  has  consistently  used  it  ever  since.  The 
Brothers  Dennis  adopted  it  for  3  1/2-ton  commercial  vehicles 
and  subsequently  for  pleasure  cars  a  few  years  later,  and  for 
several  years  the  Lanchester  Company  and  the  Dennis  Com- 
pany were  the  only  firms  regularly  using  it.  Much  opposition 
to  its  general  adoption  existed  for  years  but  Lanchester  and 
Dennis  holding  confidently  to  their  belief  in  the  principle,  it 
began  to  be  realized  in  England  that  worm  gearing  for  final 
drives  of  automobiles  was  a  serious  competitor  of  bevel  gears 
and  could  be  considered  as  having  passed  the  experimental 
stage.  Gradually  other  builders  took  it  up  and  made  it  success- 
ful in  wider  use  until  finally  the  seal  of  approval  was  set  by  the 
enormous  fleet  of  vehicles  built  and  owned  by  the  London 
General  Omnibus  Company,  all  driven  through  this  type  of 
gear. 

A  few  attempts,  distinguished  by  lack  of  self-confidence,  were 
made  to  use  it  in  the  United  States,  with  little  success  on  the 
part  of  manufacturers  or  enthusiasm  shown  by  the  public, 
until  the  author,  who  had  experimented  with  it  for  two  years 
in  England,  was  privileged  to  introduce  it  into  the  United 
States,  in  the  year  1911,  on  an  extensive  scale,  as  part  of  the 
regular  product  of  The  Pierce- Arrow  Motor  Car  Company,  and 
its  use  is  now  very  general  in  England,  on  the  Continent  of 
Europe,  and  is  rapidly  extending  in  America. 


INTRODUCTORY  3 

Certain  obvious  advantages  have  contributed  to  its  adoption 
in  automobiles  of  all  kinds  from  the  lightest  pleasure  cars  to  the 
heaviest  commercial  vehicles,  and,  while  there  is  everything  to 
be  said  in  its  favor,  its  design,  has  been  surrounded  with  so 
much  mystery  that  many  have  been  deterred  from  adopting  it. 
It  is  true  that  some  writers  have  attempted  to  ventilate  the 
subject,  and  some  mention  of  it  will  be  found  in  most  works 
devoted  to  gearing,  but  the  author  has  searched  in  vain  for  any 
book  treating  the  whole  subject  comprehensively.  Literature 
specially  devoted  to  worm  gears  for  automobiles  is,  indeed, 
practically  non-existent,  and,  at  the  best,  consists  of  occasional 
contributions  to  technical  magazines.  Most  of  such  articles 
omit  altogether  the  fundamental  principles,  a  careful  study 
of  which  is  a  necessity  if  successful  result  is  to  be  obtained  by 
other  methods  than  mere  chance. 

Much  opposition  to  the  use  of  worm  gearing  has  been  due  to 
a  misconception,  for  which  practically  all  the  text-books  are 
responsible,  that  it  is  inefficient,  and  (for  automobile  work) 
irreversible;  it  will  presently  be  shown  that  it  is  neither,  but 
mention  may  be  made  here  of  a  remarkable  kind  of  gear  which 
appeared  in  the  closing  years  of  the  nineteenth  century.  It 
was  known  as  the  "Globoid  Gear"  and  the  supposed  excessive 
friction  of  an  ordinary  worm  was  sought  to  be  overcome  by  an 
arrangement  of  conical  rollers  mounted  upon  radial  pins 
around  the  periphery  of  the  worm  wheel;  these  rollers  were 
meshed  by  a  worm  of  very  large  pitch  and  the  gear  certainly 
worked  freely  in  both  directions;  beyond  that,  it  was  very 
cumbersome,  and  after  a  little  use  noisy.  It  was  never  used  in 
automobiles,  and  is  only  referred  to  here  to  show  how  the  action 
of  this  type  of  gearing  has  been  misrepresented — even  to  the 
extent  of  inducing  inventors  to  develop  complicated  mechanisms 
to  overcome  supposed  faults  which  do  not  in  reality  exist. 

Since  the  early  days  of  experiments  so  much  actual  experience 
in  daily  use  with  worm  gears  of  all  sizes  has  been  possible  that 
it  can  now  be  said  that  something  approaching  finality  has 
been  reached,  and  the  art  of  making  worm  gearing  has  been 
placed  upon  a  substantial  foundation. 

Such  is,  in  outline,  the  history  of  this  most  interesting  me- 


4  WORM  GEARS 

chanical  device,  and  with  this  very  brief  glance  at  its  early  use, 
we  will  pass  to  the  consideration  of  the  principles  of  its  design 
and  construction,  dealing  in  as  progressive  a  manner  as  possible 
with  a  subject  which  is  necessarily  of  a  somewhat  complex 
nature. 


CHAPTER  II 

CHOICE  OF  MATERIALS  AND  METHODS  OF 
MANUFACTURE 

At  different  times,  many  combinations  of  metals  have  been 
proposed  for  making  worm  gears.  Some  kind  of  bronze  for  the 
wheel  and  steel  for  the  worm  was  adopted  as  the  most  natural 
combination  for  two  sliding  surfaces  working  together ;  the  first 
and  most  obvious  improvement  being  to  case-harden  the  worm 
to  minimize  wear.  Rise  in  the  cost  of  copper  and  tin  suggested 
the  use  of  other  metals  for  the  wheel,  and  cast  iron  of  various 
grades,  mild  steel,  and  even  case-hardened  steel  were  tried  for 
the  wheels,  all  in  combinations  with  and  without  a  hardened 
worm.  For  many  years,  bronze  has  been  used  for  the  worm 
wheels  of  elevators  driven  by  a  steel  worm  and  it  is  now  gen- 
erally recognized  that  case-hardened  steel  for  the  worm  and 
phosphor  bronze  for  the  wheel  cannot  be  surpassed.  It  is 
interesting  to  note  here  that  manganese  bronze  is  unsuitable, 
presumably  on  account  of  its  hardness.  The  bronze  must  be 
very  homogeneous  and  close  grained,  and  great  care  must  be 
taken  to  ensure  uniformity  of  mixing,  melting  and  pouring,  in 
order  to  ensure  uniform  results.  The  writer  recommends  a 
mixture  of  the  following  proportions : 

Copper 89 

Phosphorous 1 

Tin 10 

This  will  have  an  elastic  resistance  to  crushing  amounting  to 
22,000  Ib.  per  square  inch,  and  to  penetration,  under  a  hardened 
steel  point  of  .125  sq.  in.  area,  of  5000  Ib.  On  the  Shore 
scleroscope,  it  will  give  a  hardness  reading  of  15  to  20.  Its 
tensile  strength  will  amount  to  35,000  Ib.  per  square  inch. 
With  such  a  metal,  a  safe  working  stress  of  7000  Ib.  per  square 
inch  may  be  permitted 


6  WORM  GEARS 

For  the  worm,  seeing  that  its  shape  conduces  to  great  strength, 
no  unusual  strength  is  required  in  the  material,  and  almost  any 
low  carbon  case-hardening  steel,  which  can  be  heat  treated 
without  serious  distortion,  may  be  employed.  Steel  for  this 
purpose  should  contain  3  to  3.5  per  cent,  nickel  and  .16  to  .18 
carbon.  In  its  untreated  state,  it  will  give  approximately  the 
following  physical  tests,  on  a  standard  2-in.  specimen. 

Elastic  limit 62,000  Ib.  per  square  inch. 

Maximum  strength 82,000  Ib.  per  square  inch. 

Elongation 30  per  cent. 

Reduction 60  per  cent. 

After  machining  to  size  with  an  allowance  of  say  .003  in. 
for  grinding,  it  can  be  subjected  to  the  following  heat  treat- 
ment. Pack  in  carbonizing  material  and  maintain  for  eight 
hours  at  a  temperature  of  1600  to  1650°  F.,  allow  to  cool  in  the 
carbon,  reheat  to  1350°  F.  and  quench  in  oil. 

The  above  may,  of  course,  be  varied  at  the  discretion  of  the 
designer,  the  object  being  to  obtain  a  thick  casing  of  a  hardness 
which  will  indicate  from  60  to  70  on  the  scleroscope. 

After  hardening,  the  worm  may  be  finished  to  required  size 
by  grinding  which  will,  if  properly  performed  in  a  suitable 
machine,  provide  a  worm  accurate  upon  all  threads  and  with  a 
perfect  and  lasting  working  surface. 

With  regard  to  the  method  of  manufacture  of  gears,  the  cast- 
ing of  the  wheel  should  be  rough  turned  and  roughly  gashed  in 
a  milling  machine,  after  which  it  is  strongly  recommended  that 
it  be  set  on  one  side  to  season.  Being  necessarily  of  consider- 
able mass,  the  surface  tension  set  up  during  cooling,  on  being 
released  by  machining  will  cause  some  distortion,  and  seasoning 
for  a  period  of  some  weeks  will  allow  the  metal  to  assume  a 
permanent  form  which  it  may  be  relied  upon  to  maintain.  The 
wheel  is  then  turned  to  exact  size  and  the  teeth  cut  by  means  of 
hobbing.  It  is  safe  to  say  that  few  makers  have,  until  recently, 
appreciated  the  necessity  for  employing  extra  rigid  machinery 
for  this  purpose  without  which  the  highest  class  of  work  is 
impossible.  By  the  older  method  of  hobbing,  a  parallel  hob 
was  fed  radially  into  the  wheel,  but  in  more  modern  methods 
a  taper  hob  is  fed  at  fixed  centers  tangentially  to  the  wheel  it 


CHOICE  OF  MATERIALS  7 

is  cutting;  the  reason  for  this  is  obvious — by  such  a  method 
alone  can  the  centers  be  preserved  at  the  required  distance  and 
inter  changeability  of  similar  gears  be  assured.  It  is  not  possible 
to  obtain  the  same  finish  on  the  worm  wheel  as  in  the  case  of  the 
worm,  but  some  makers  have  been  particularly  successful  in 
this  respect,  finishing  the  worm  wheel  with  a  special  kind  of  hob 
which  leaves  an  extremely  accurate  surface  which  immediately 
beds  to  the  worm. 

In  the  case  of  the  worm,  it  is  an  advantage  to  rough  turn  it 
and  rough  cut  the  thread,  and  then  anneal  it,  to  remove  internal 
strains  and  subsequent  distortion,  and  while  in  some  steels  it  is 
not  necessary,  it  is  a  practice  which  may  be  recommended.  The 
cutting  of  the  worm  may  be  performed  in  a  lathe  with  a  single 
point  tool  accurately  ground  to  the  required  form,  and  for 
experimental  work,  saving  the  expense  of  cutters,  this  has  its 
uses;  the  better  method,  however,  is  of  course,  to  mill  the  thread 
in  a  thread  milling  machine,  and  special  machines  are  generally 
used  for  repetition  work.  In  the  case  of  the  hollow  or  Hindley 
worm,  as  used  by  Mr.  Lanchester  and  others  who  have  followed 
his  example,  a  special  hobbing  machine  is  used  for  cutting  the 
worm,  to  which  reference  will  be  made  again  later.  Such  a 
worm  cannot  be  ground,  and,  after  hardening,  must  be  cleaned 
up  on  its  working  surfaces  by  hand.  At  the  time  of  writing, 
the  author  is  not  aware  of  any  parallel  worms  having  been 
hobbed,  but  it  seems  likely  that  there  may  be  developments  in 
this  direction  in  the  future.  Whatever  method  is  adopted, 
however,  the  greatest  possible  accuracy  must  be  insisted  upon 
in  the  case  of  both  the  worm  and  the  wheel,  the  high  efficiency 
of  modern  worm  gears  being  almost  entirely  due  to  extreme  care 
in  this  respect. 


CHAPTER  III 

DEFINITIONS  AND  SYMBOLS 

A  worm  gear  may  be  defined  as  a  spur  wheel  which  is  rotated 
by  an  endless  rack,  the  teeth  of  which  are  successively  pressed 
against  the  teeth  of  the  wheel.  By  making  the  rack  teeth  in 
the  form  of  a  spiral  and  rotating  it  upon  its  axis  (sloping  the 
wheel  teeth  to  a  corresponding  angle),  the  effect  of  an  infinitely 
long  rack  is  obtained. 

Such  a  rack  is  called  a  parallel  worm.  By  revolving  the  worm 
wheel,  the  teeth  of  the  rack  may  be  caused  to  move  along,  that 
is  to  say,  the  worm  will  itself  commence  to  rotate,  the  relative 
motions  being  thus  convertible.  The  worm  is  then  said  to  be 
''reversible/7  the  amount  of  reversibility  depending  upon 
certain  fixed  principles  which  will  be  later  discussed. 

In  order  to  obtain  a  greater  area  of  surface  contact  between 
the  wheel  and  the  rack,  the  latter  may  be  curved  to  partially 
embrace  the  wheel,  the  form  of  the  worm  then  partaking  of 
that  of  an  hour  glass.  This  form  is  usually  known  as  the  Hind- 
ley  worm,  from  its  inventor,  and  has  been  brought  to  a  high 
state  of  perfection  for  use  in  automobiles  by  Mr.  Lanchester. 
As  the  central  portion  of  a  Hindley  worm  exactly  corresponds 
in  its  action  with  that  of  a  parallel  worm,  the  same  calculations 
are  applicable  to  either,  with  the  exception  of  such  formulae  as 
refer  to  the  number  of  teeth  in  engagement  at  one  time  and 
consequently  the  length  of  the  worm;  these  differences  will  be 
treated  in  their  proper  place. 

We  will  now  define  clearly  the  following  expressions  to  which 
frequent  allusion  will  have  to  be  made. 

Gear  Ratio. — As  in  any  other  system  of  gearing,  the  ratio 
of  the  speeds  of  revolution  which  one  member  will  make  when 
meshing  with  the  other  is  called  the  gear  ratio;  thus,  with  a  3  to 

8 


DEFINITIONS  AND  SYMBOLS  9 

1  ratio,  it  is  to  be  understood  that  the  worm  will  make  exactly 
three  revolutions  to  one  revolution  of  the  worm  wheel.  Unlike 
spur  or  bevel  gears,  however,  the  ratio  is  not  primarily  depend- 
ent on  the  number  of  threads  of  the  worm,  but  involves  the 
question  of  lead. 

Lead. — This  may  be  defined  as  the  distance  traveled  along 
the  axis  by  one  thread  in  one  revolution;  it  is  measured  in 
inches.  How  this  determines  the  gear  ratio  may  be  made  clear 
by  the  following  example.  If  the  worm  wheel  be  30  in.  in 
circumference  and  the  worm  has  a  lead  of  6  in.,  the  gear  ratio 

30 

will  be  -  —  =  5  to  1.  This  may  be  further  defined  as  the  dimen- 
sional ratio,  obtained  by  dividing  the  circumference  of  the  wheel 
by  the  lead,  so  that  we  may  say  as  a  starting-point  that 

wheel  circumference 

-  =  gear  ratio  (1) 

worm  lead 

By  increasing  the  diameter  of  the  wheel  the  number  of  teeth 
is  increased  and  the  gear  ratio  is  reduced,  while  the  converse 
of  this  is,  of  course,  true,  but  by  increasing  the  diameter  of  the 
worm  the  rubbing  velocity  is  increased. 

By  increasing  the  lead  angle,  the  gear  ratio  is  reduced;  the 
circumferential  pitch  being  first  determined  and  the  number  of 
wheel  teeth,  it  follows  that  a  single-threaded  worm  will  give  a 
reduction  of  1  to  n  where  n  =  the  number  of  wheel  teeth.  A 
double-threaded  worm  will  have  a  reduction  of  2  to  n  and  so  on, 
but  this  relationship  is  not  primarily  due  to  the  number  of 
threads  in  the  worm  but  to  the  lead  of  the  thread  which  may  be 
extended  to  include  at  each  revolution  one,  two,  or  more  teeth. 
Thus,  in  a  worm  having  five  threads,  each  thread  will  make  a 
complete  engagement  with  only  every  fifth  tooth  in  the  wheel, 
and  the  other  four  threads  will  pick  up  the  intervening  teeth 
respectively. 

Pitch  Line  of  Wheel. — Properly  speaking  a  worm  wheel  has 
no  pitch  line  at  all,  that  is  to  say,  in  the  sense  of  the  two  rolling 
circles  of  spur  or  bevel  gears;  it  is,  however,  a  convenient  expres- 
sion for  a  circle  corresponding  to  the  mean  effective  diameter  of 
the  wheel  and  must  be  so  understood. 


10  WORM  GEARS 

Pitch  Line  of  Worm. — This  also  has  only  an  hypothetical 
existence,  but  may  be  defined  as  the  circle  described  about  the 
axis  of  the  worm  which  touches,  at  right  angles,  the  pitch  circle 
of  the  wheel.  All  subsequent  references  to  "Pitch"  whether 
applied  to  the  worm  or  wheel  must  be  understood  to  be  meas- 
ured upon  these  circles. 

Circular  Pitch  of  Wheel. — The  distance  in  inches,  or  fractions 
of  an  inch,  between  the  centers  of  the  wheel  teeth  measured 
along  the  pitch  line. 

Axial  Pitch  of  Worm. — The  distance  in  inches,  or  fractions  of 
an  inch,  between  the  centers  of  adjacent  worm  threads,  meas- 
ured along  the  pitch  line,  parallel  to  the  axis  of  the  worm. 

Note. — In  a  worm  having  more  than  one  thread,  this  is  not 
the  same  as  the  lead  of  the  worm.  In  fact,  the  lead  is  the  prod- 
uct of  the  number  of  threads  or  starts  multiplied  by  the 
axial  pitch.  The  axial  pitch  of  the  worm  is  always  equal  to  the 
circular  pitch  of  the  wheel. 

Normal  Pitch  of  Worm  and  Wheel. — This  is  the  distance 
between  the  centers  of  two  adjacent  threads  or  teeth  measured 
normally  or  at  right  angles  to  their  faces  along  the  pitch  line. 
Owing  to  the  inclination  of  the  threads  or  teeth,  this  distance 
is  always  less  than  the  circular  or  the  axial  pitch. 

Circumferential  Pitch  of  Worm. — The  distance  along  the 
pitch  line  between  two  adjacent  worm  threads,  measured  cir- 
cumferentially  round  the  worm. 

Lead  Angle. — The  angle  formed  by  the  thread  of  the  worm 
with  a  line  drawn  at  right  angles  to  the  axis  of  the  worm.  If 
the  spiral  formed  by  the  threads  were  unwrapped  from  the 
worm,  it  would  form  an  inclined  plane  having  an  inclination 
equal  to  the  lead  angle. 

Addendum. — The  height  of  wheel  tooth  or  worm  thread 
outside  pitch  line. 

Dedendum. — The  depth  of  wheel  tooth  or  worm  thread  inside 
pitch  line. 

Length  of  Worm. — The  length  of  the  cylinder  bounded  by  the 
pitch  line  of  the  worm.  If  two  radii  be  drawn  from  the  center 
to  the  circumference  of  the  pitch  line  of  the  wheel,  so  that  each 
touches  the  extremity  of  this  cylinder,  they  will  make  an  angle 


DEFINITIONS  AND  SYMBOLS  11 

which  is  subtended  by  the  cylinder  of  the  worm  pitch  line,  and 
the  length  of  this  cylinder  is  the  chord  of  this  angle. 

Note. — The  length  of  the  worm  has  nothing  to  do  with  the 
the  lead  of  the  worm. 

Subtended  Angle  of  Worm. — If  two  radii  be  drawn  from  the 
axis  to  the  circumference  of  the  pitch  circle  of  the  worm,  they 
will  form  an  angle  which  is  filled  by  the  teeth  of  the  worm 
wheel.  This  is  the  subtended  angle  of  the  worm,  and  is  the  only 
manner  in  which  the  effective  width  of  the  worm  wheel  can  be 
measured.  The  determination  of  this  angle  has  a  very  impor- 
tant bearing  upon  the  performance  of  the  gear. 

Included  Angle  of  Thread. — In  a  V-threaded  worm,  the  angle 
made  by  the  inclination  of  the  faces  of  the  V  is  the  included 
angle. 

Pressure  Angle. — Half  the  included  angle. 

Note. — Both  the  included  angle  and  the  pressure  angle  may 
be  measured  normally  or  in  the  direction  of  the  axis;  unless 
otherwise  specified,  however,  the  axial  angles  are  always  to  be 
understood  to  be  implied. 

The  following  system  of  symbols  will  be  employed  through- 
out this  work. 

G  =Gear  ratio. 

D  = Pitch  line  diameter  of  worm  wheel  in  inches. 

d  =  pitch  line  diameter  of  worm  in  inches. 

L  =Lead  of  worm  in  inches. 

I  =  Length  of  thread  per  revolution  in  inches. 

N  =  Number  of  teeth  in  worm  wheel. 

n  =  Number  of  teeth  in  worm. 

R  =  Pitch  line  radius  of  worm  wheel  in  inches. 

r  =Pitch  line  radius  of  worm  in  inches. 

R'  =  Radius  of  road  wheel  in  inches. 

#"  =  Extreme  radius  of  worm  wheel  on  plane  of  worm  axis. 

v  =  Rubbing  velocity  in  feet  per  second. 

T  =  Torque  in  pounds  at  pitch  line  of  worm. 

p  =  Tangential  pressure  at  pitch  line  in  pounds. 

p'  =  Normal  tooth  pressure. 

P  =  Circular  pitch  of  worm  wheel  and  axial  pitch  of  worm. 


12  WORM  GEARS 

P'  =  Normal  pitch  of  worm  threads  and  wheel  teeth. 

P"  =  Circumferential  pitch  of  worm  threads. 

/    =  Force  separating  worm  and  wheel. 

w    =  Weight  at  rear  axle  in  pounds  on  the  ground. 

x    =Safe  load  on  tooth  of  worm  wheel,  in  pounds. 

g    =  Length  of  worm  in  inches. 

nf  =  Number  of  teeth  in  contact. 

a    =Lead  angle. 

6    =  Included  angle  of  worm  thread  (axial). 

W   =  Normal  included  angle  of  thread. 

a    =  Stress  permissible  in  wheel  teeth  in  pounds  per  square 

inch. 

1C   ^Gliding  angle. 
<p    =  Angle  of  friction  (tan  <p  =  fi). 
jj.    =  Coefficient  of  friction. 
T)    =  Mechanical  efficiency. 
/?    =  Angle  of  worm  subtended  by  wheel. 

n 

CD    =  Pressure  angle  =  -  * 

X     =  Length  of  tooth. 

p 

T     =  Thickness  of  worm  thread  at  pitch  line  =  — ' 

it 

K  =Dedendum. 

o  =  Addendum. 

p  =  Re  volutions  of  worm  per  minute. 

^  —  Angle  of  worm  wheel  subtended  by  worm. 


CHAPTER  IV 

PRELIMINARY  PROPORTIONS 

Certain  data  are  required  before  the  design  can  be  attempted. 
It  is  usually  necessary  to  know  the  following: 

(a)  The  gear  ratio. 

(b)  The  maximum  torque  of  the  motor . 

(c)  The  lowest  gear  ratio  in  the  transmission. 

Note. — The  torque  delivered  to  the  worm  will  be  the  maxi- 
mum torque  of  the  motor  multiplied  by  the  lowest  gear  ratio 
(generally  the  reverse  gear)  less  the  drop  in  mechanical  efficiency 
due  to  the  gears  and  bearings. 

(d)  The  maximum  speed  of  the  motor  and  especially  the 
speed  at  which  the  maximum  torque  is  given  off.     In  an  internal 
combustion  motor  the  speed  of  maximum  torque  is  always  less 
than  the  speed  of  maximum  power. 

(e)  The  total  weight  on  the  ground  at  the  rear  axle. 

(f)  The  diameter  of  the  road  wheels. 

The  first  thing  to  settle  is  the  circular  pitch,  and  while  a  great 
deal  of  latitude  is  permissible  in  this,  some  limitations  must  be 
set  to  it.  It  will  be  shown  that  the  strength  of  the  wheel  teeth 
varies  directly  with  the  circular  pitch;  the  smallest  pitch  even 
in  the  lightest  cars  which  can  be  employed  is  13/16  in.,  since 
less  than  this  gives  very  little  margin  for  wear  or  for  hidden 
defects  in  the  bronze  of  the  worm  wheel,  but  it  is  clearly  im- 
possible to  increase  the  circular  pitch  in  direct  proportion  to 
the  power  to  be  transmitted.  The  pitch  diameter  of  the  worm 
naturally  affects  the  torque  loads  on  the  teeth,  and  it  will  be 
necessary  to  increase  the  worm  diameter  in  some  proportion  to 
the  power,  so  that  it  seems  that  some  increase  of  the  circular 
pitch  with  the  diameter  of  the  worm  is  a  natural  sequence. 
Very  coarse  pitches  are  in  use  notably  in  the  gears  adopted  as 
standard  in  the  vehicles  of  the  London  General  Omnibus 

13 


14  WORM  GEARS 

Company  which  are  in  the  neighborhood  of  1  7/8  in.  but  there 
is  no  advantage  in  going  above  11/4  in.  The  author  has 
employed  1  1/8  in.  in  gears  transmitting  over  90  H. P.  through 
a  worm  with  a  pitch  line  diameter  of  no  less  than  4  3/4  in.  This 
in  heavy  pleasure  cars  has  given  perfectly  satisfactory  results. 
The  disadvantage  of  a  coarse  pitch  is  this,  when  at  work,  the 
surfaces  of  the  threads  and  the  teeth  are  separated  by  a  film  of 
oil,  the  coarser  the  pitch,  the  fewer  the  number  of  teeth  in  con- 
tact, consequently  the  tangential  pressure  between  the  worm 
and  the  wheel  must  be  carried  on  fewer  films  of  oil,  that  is  to 
say  the  specific  tooth  pressure  will  be  greater  and  this  may  rise 
to  a  point  where  the  oil  is  squeezed  out  and  lubrication  becomes 
ineffective;  a  fine  pitch  is  therefore  always  better  than  a  coarse 
one,  other  things  being  equal. 

There  is  another  aspect  to  this,  the  worm  must  always  be 
rigid  beyond  the  possibility  of  springing  between  its  journals, 
with  a  very  coarse  pitch  so  much  stock  may  be  removed  from 
the  body  of  metal  as  to  seriously  reduce  its  transverse  strength, 
moreover  some  slight  angular  torsion  must  theoretically  occur 
in  the  worm  itself,  and  if  the  cross-section  of  the  threads  is 
considerably  greater  than  that  of  the  central  spindle,  there  is  a 
tendency  for  the  threads  to  unwrap  and  tear  away  from  the 
body  of  the  structure;  a  similar  result  being  very  familiar  to 
automobile  engineers  in  the  case  of  heavily  stressed  splined 
shafts,  which,  if  the  splines  be  very  coarse  and  rigid  in  com- 
parison with  the  rest  of  the  shaft,  will  tear  asunder  like  the 
quarters  of  an  orange. 

All  these  considerations  therefore  point  to  the  desirability  of 
employing  a  comparatively  fine  pitch,  and  in  order  to  ensure 
stiffness  in  the  spindle  and  to  leave  sufficient  metal  in  this,  a 
pitch  diameter  less  than  2  in.  is  seldom  required. 

Based  upon  examples  from  actual  practice,  the  following 
pitches  as  shown  in  Table  1,  will  be  found  satisfactory  for 
different  horse-powers: 

These  are  shown  diagrammatically  in  Fig.  1,  and  while  it 
may  be  expedient  to  depart  slightly  from  these  relationships  to 
suit  special  features  of  axle  design,  they  should  be  as  closely  as 
possible  adhered  to. 


PRELIMINARY  PROPORTIONS 


15 


TABLE  I 


H.P. 

Circ.  pitch 

Pitch  dia. 

15 

if  in. 

2^  in. 

25 

*in. 

2|  in. 

50 

1    in. 

3^  in. 

75 

If  in. 

4|  in. 

100 

H  in. 

5    in. 

5.5 
5.0 
4.5" 
4.0" 
3.5 
3.0 
2.5" 
2.0" 
1.5 
1.0" 
0.5 
0 


10 


20 


40          50          60  70 

Horse  Power 

FIG.  1. 


90         100 


Having  selected  from  this  table  the  suitable  pitch  and  worm 
diameter  for  a  given  horse-power,  the  next  thing  to  determine 
is  the  number  of  wheel  teeth  and  worm  threads  to  give  the 


16 


WORM  GEARS 


required  ratio.  Seeing  that  a  comparatively  small  pitch  has 
been  selected  for  all  wheels,  it  is  evident  that  a  relatively  greater 
number  of  worm  threads  is  possible. 


DETERMINATION  OF  WORM  WHEEL  DIAMETER 

This  will  naturally  be  preferred  as  small  as  possible  but  it 
must  not  be  reduced  at  the  expense  of  excessive  tangential 
pressure,  and  the  number  of  teeth  should  not  be  less  than  24  or 
more  than  40.  The  writer  has  found  that  almost  any  conditions 
can  be  satisfied  with  between  30  and  40  teeth,  bearing  in  mind 
that  a  very  heavy  vehicle  requires  a  larger  wheel  than  a  light 
one,  but  the  heaviest  commercial  vehicle  need  not  have  more 
than  40,  and  questions  of  road  clearances  under  the  axle  will  to 
a  great  extent  determine  the  diameter  which  must  always 
remain  a  matter  of  judgment  on  the  part  of  the  designer. 

The  following  proportions  of  teeth  are  recommended: 


FIG.  2. 


Addendum, 
Dedendum, 


.3183P 
.3683P 


It  has  been  stated  that  the  pitch  of  the  worm  can  be  measured 
in  three  ways  (Fig.  2) : 

Axially  =P 

Normally  =  Pf 

Circumf  erentially  =  P" . 

and  these  are  inter-related  as  follows: 


PRELIMINARY  PROPORTIONS  17 

P'  =  P  cos  a  } 

P"  =  p>  cosec  a  [ 

Therefore,       P"  =  P  cos  a  cosec  a 


P"  =  P  cot  a 

It  follows  from  this  that  the  number  of  threads  on  a  worm 
has  the  following  relation  to  its  pitch  line  diameter. 

Ttd  =  n  P  cot  a  (3) 

(4) 


P  cot  a 

and  to  find  the  number  of  threads,  it  is  first  necessary  to  know 
the  value  of  the  lead  angle  a  which  again  depends  upon  the 
lead  L,  and  this  in  its  turn  upon  the  gear  ratio  required. 
The  gear  ratio  G  has  the  following  value 

G  =  ~  (5) 

hence  L  =  *D  (6) 

and  the  lead  angle  a  is  such  that 

tan  a  =  ±  (7) 

Substituting  from  equation  (6) 

tan  a  =  -— 
nd 

tan  a  =  — 


D 

tan  a  =  -^  (8) 

tttr 

trom  which  a  is  at  once  found,  and  from  this  the  value  of  n 
(equation  4). 

DETERMINATION  OF  LENGTH  OF  WORM  AND  NUMBER  OF 
TEETH  IN  CONTACT 

The  worm  and  worm  wheel  may  be  regarded  as  two  cylinders 
which  cut  one  another,  their  axes  being  at  right  angles. 


18 


WORM  GEARS 


In  Fig.  3,  which  is  a  section  through  the  axis  of  the  worm,  the 
line  dc  represents  the  limiting  surface  of  the  worm;  o  is  the 
addendum  of  the  teeth. 

R",  the  radius  of  the  limiting  circle  of  the  worm  wheel,  is 
equal  to  R  +o. 


FIG.  3. 
The  total  height  of  the  wheel  teeth  cutting  the  cylinder  of  the 

<N 

worm  is  o  +o,  and  this  is  also  the  versin  of  the  angle  ~. 

Therefore,  2X° 

Again, 


--=  versin  - 


=  2  sn- 


/.the  length  of  dc  =  2  R"  sin 


9 


(9) 


Since  the  teeth  of  the  wheel  are  hobbed,  it  follows  that  every 
space  between  the  worm  threads  has  a  wheel  tooth  corre- 
sponding to  it  and  some  portion  of  this  is  in  contact  with  a 
thread  of  the  worm. 


The  number  of  threads  is  ~ 


(10) 


PRELIMINARY  PROPORTIONS  19 

and  this  is  the  number  of  teeth  in  contact  at  all  times.     Let 
this  =n'. 


Then,  n'  =  p  (11) 

Substituting  for  g, 

(12) 


Up  to  this  point  we  have  considered  the  method  of  determin- 
ing the  following: 

Pitch  of  the  wheel  teeth, 

Diameter  of  worm, 

Diameter  of  worm  wheel, 

Lead  angle, 

Number  of  worm  threads, 

Length  of  worm. 

The  centers  of  the  worm  and  wheel  are  evidently  expressed 
by  the  following 

c^-  (13) 

and  having  thus  settled  the  leading  dimensions,  we  are  in  a 

position  to  investigate  the  correct  form  of  thread,  .and  the 

required  strength  of  the  various  parts  to  withstand  the  stresses 

to  which  they  will  be  subjected. 

It  is  convenient  at  this  point  to  introduce  a  number  of 

formula  which  will  frequently  have  to  be  referred  to  in  the  rest 

of  this  investigation. 

Lead  of  worm  L  =  Pn  (14) 

Length    of   thread    per   revolution    =l  =  ndseca  (15) 

hence,  rubbing  velocity  of  worm 

.p.m.  . 

ft-    er  second- 


2X6C 


Lead  angle  a.  may  be  found  from 


tan  «=—  y  (17) 


20  WORM  GEARS 

Whence  it  follows  that  the  angle  of  lead  is  not  affected  by  the 
number  of  teeth  in  the  wheel. 

The  rubbing  velocity  v  is  given  by  some  writers  as  being  the 
circumferential  speed  of  the  worm;  this  is  incorrect  and  would 
only  be  true  of  a  worm  having  its  lead  angle  =  o;  it  is  of  course 
dependent  upon  the  length  of  thread  (per  revolution  of  worm) 
measured  around  the  spiral  and  equation  (15)  gives  the  correct 
value.  The  manner  in  which  the  rubbing  velocity  is  affected 
by  the  diameter  of  the  worm  may  be  illustrated  in  the  follow- 
ing manner. 

It  is  evident  that  a  given  lead  may  be  obtained  by  a  large 
diameter  worm  with  a  small  lead  angle,  or  a  small  worm  with  a 
very  quick  angle,  but  that  similar  results  will  give  very  different 
conditions  of  working  is  apparent  from  the  following  examples. 
Take  the  case  of  three  worms  having  each  a  linear  pitch  of  5  in., 
but  with  pitch  line  diameters  of  (a)  3  in.,  (6)  5  in.,  and  (c)  7  in. 
respectively. 

It  follows  from  equation  (17)  that  the  lead  angles  of  such 
worms  will  be  as  follows: 

(a)  tan    a=--  =    -z=-53l    whence    a  =28° 


5         5 

(b)  tan    «  =—=---—-  =.318  whence  a=17°39' 

D7T       JLO.  /  U 

5         5 

(c)  tan    «=-  =  —  -gg=.2274  whence  a=12°  49' 

and  the  rubbing  velocities  from  equation  (16)  will  be  at  1000 
revolutions  per  minute. 

,  .  1.132X9.42X1000 

(a)  -  -   =14.8  ft.  per  second. 


1.049X15.7X1000 

—  =  22.9  it.  per  second. 


v  1.025X21.99X1000 
(c)  -         ~~720~        —=31.3  ft.  per  second. 

The  immediate  effect  of  this  is  at  once  apparent,  since  to 
revolve  the  worm  wheel  for  one  second  of  time  the  surfaces*of 


PRELIMINARY  PROPORTIONS  21 

the  three  worms  referred  to  will  have  to  travel  through  these 
distances  in  order  each  to  revolve  the  worm  wheel  through  the 
same  amount  and,  as  the  lost  work  of  the  worm  per  revolution 
is  represented  bylX^Xp'  where  I  is  length  of  thread  per  revolu- 
tion, fj.  is  coefficient  of  friction,  and  p'  the  tooth  pressure,  it 
follows  that  the  losses  will  be  directly  proportional  to  the  length 
of  thread,  that  is,  the  rubbing  speed. 


CHAPTER  V 

PRESSURE  ANGLE  AND  FORM  OF  THREAD 

It  has  already  been  remarked  that  much  misconception 
formerly  existed  as  to  the  possibility  of  any  worm  gear  being 
reversible,  that  is  to  say,  that  an  automobile  so  fitted  would  be 
unable  to  "  coast."  In  this  connection,  the  form  of  thread  is 
of  the  utmost  importance  and  by  its  manipulation,  we  can 
provide  within  practical  limits  perfect  reversibility  with  any 
lead  angle  of  ordinary  dimensions. 

Let  it  always  be  remembered  that  a  section  through  the  worm 
on  a  plane  including  its  axis,  will  represent  it  as  a  rack,  and  the 
worm  wheel  as  a  pinion  rolling  upon  it.  It  is  not  necessary  to 
state  that  if  the  sides  of  the  teeth  of  the  rack  be  straight,  the 
teeth  of  the  pinion  will  be  of  an  involute  form;  moreover,  if  the 
wheel  were  of  very  large  diameter,  the  sides  of  the  teeth  of  the 
rack  could  be  square  or  nearly  so.  The  size  of  the  wheel,  how- 
ever, in  an  automobile  is  necessarily  very  limited,  and  conse- 
quently if  the  rack  teeth  or  worm  threads  were  made  square,  a 
great  amount  of  undercutting  would  ensue  in  the  generation  of 
the  wheel  teeth,  and  to  avoid  this,  the  threads  of  the  worm 
must  be  cut  away  in  the  form  of  a  V,  and  this  brings  us  to  the 
question  of  the  pressure  angle. 

In  their  " Practical  Treatise  on  Gearing/'  Messrs.  Brown  & 
Sharpe  recommend  a  pressure  angle  of  14  1/2  degrees,  that  is 
to  say,  the  threads  of  the  teeth  will  have  an  included  angle  of 
29  degrees;  it  is,  however,  pointed  out  in  the  same  work,  that 
interference  of  the  rack  teeth  will  begin  in  wheels  of  31  teeth  if 
this  angle  be  adopted.  Something  larger  than  this  will  there- 
fore have  to  be  selected.  There  is,  however,  another  aspect  to 
this,  and  in  it  is  involved  the  question  of  reversibility. 

It  must  be  remembered  that  the  surface  of  the  worm  thread 
is  constantly  slipping  over  the  faces  of  the  wheel  teeth,  and  the 

22 


PRESSURE  ANGLE  AND  FORM  OF  THREAD   23 

direction  in  which  this  slipping  takes  place  is  along  a  plane 
mutually  tangent  to  the  face  of  the  thread,  and  the  face  of  the 
wheel  teeth.  It  is  evident  that  if  this  gliding  angle  be  45 
degrees,  it  will  be  immaterial  whether  the  worm  drives  the  wheel 
or  the  wheel  the  worm;  with  a  square  thread,  however,  such  a 
gliding  angle  is  only  possible  if  the  proportions  of  the  worm  are 
such  that  they  are  satisfied  by  the  equation 

7id  =  L  (18) 

The  gear  ratio  and  the  dimensions  of  the  worm  and  wheel  may, 
and  generally  will,  make  this  impossible. 

In  Fig.  4,  let  a  b  c  be  a  triangle  where  a  c  equals  the  lead  and 
b  c  is  equal  to  the  circumference  of  the  pitch  line  of  the  worm. 
Then  the  angle  a  b  c  equals  the  lead  angle  of  the  worm.  The 


working  surface  of  the  thread  is  shown  in  perspective  by  the 
plane  a  e  d  b  and,  in  a  square-threaded  worm,  the  angle  a  b  c  is 
the  gliding  angle  of  the  rubbing  surfaces.  As  drawn,  however, 
this  angle  is  much  less  than  45  degrees  and  such  a  gear  cannot 
be  reversible,  except  to  a  slight  degree.  Now  suppose  the  plane 
a  e  d  b  partially  rotated  about  the  line  a  b  as  an  axis  until  it 
assumes  the  position  a  g  f  b,  the  angle  which  the  plane  makes 
with  the  horizontal  plane  d  b  c  h  may  become  45  degrees  without 
any  increment  of  the  angle  a  b  c,  or  in  other  words  converting 
the  square  into  a  V  thread,  the  gliding  angle  has  been  raised  to 
45  degrees  without  increasing  the  lead  angle.  If  then  a  gliding 
angle  of  45  degrees  were  the  one  object  to  be  attained,  this 

/5 

would  be  satisfied  by  the  equation  a  +  ~  =  45  degrees  where  a 


24 


WORM  GEARS 


equals  lead  angle  and  6  equals  the  angle  included  between  the 
faces  of  the  V  thread  as  shown  in  Fig.  5,  and  any  gear  designed 


FIG.  5. 

upon  these  principles  will  be  perfectly  reversible,  provided  the 
angle  of  the  pitch  is  kept  within  reasonable  limits. 

TABLE  II.— SHOWING  REQUIRED  ANGLE  0  TO  GIVE  A  GLIDING  ANGLE 
OF  45  DEGREES 


Lead  angle,  a 

Included  angle,  6 

25 

40 

26 

38 

27 

36 

28 

34 

29 

32 

30 

30 

31 

28 

32 

26 

33 

24 

34 

22 

35 

20 

36 

18 

37 

16 

38 

14 

39 

12 

40 

10 

41 

8 

42 

6 

43 

4 

44 

2 

45 

0 

PRESSURE  ANGLE  AND  FORM  OF  THREAD      25 

Perfect  reversibility  will  be  secured  if  the  above  proportions 
are  followed;  it  has,  however,  already  been  shown  that  a  thread 
which  is  square  or  nearly  so  will  cause  destructive  undercutting 
in  the  wheel  teeth.  How  then  in  the  case  of  a  lead  angle  of 
45  degrees  is  reversibility  to  be  obtained? 

It  happens  that  with  gears  machined  by  modern  processes, 
the  coefficient  of  friction  is  very  low;  in  recent  experiments 
made  upon  the  worm  gears  for  operating  the  lock  gates  at  the 
Panama  Canal  the  coefficient  of  friction  was  so  low  that  gears 
which  were  actually  required  to  be  irreversible  were  found  to 
be  the  reverse  in  actual  practice,  and  this  fact  may  be  taken 
advantage  of  by  the  designer  of  automobile  gears  inasmuch  as 
the  angles  given  in  the  above  table  may  be  departed  from 
within  very  wide  limits  without  in  any  way  impairing  the 
reversibility. 

With  a  narrow  tooth,  such  as  is  given  with  a  14  1/2  degree 
pressure  angle,  the  difficulty  of  milling  the  worm  is  much 
increased,  in  cases  where,  as  is  usual,  the  lead  angle  is  above 
28  degrees.  By  making  a  flatter  tooth,  that  is  one  with  a 
more  open  V,  this  difficulty  to  a  great  extent  disappears,  and 
the  grinding  of  the  worm  teeth  is  also  a  far  simpler  operation. 

The  author's  practice  is  to  adopt  for  the  sake  of  uniformity  a 
universal  angle  of  60  degrees  which  greatly  simplifies  designing 
and  which  gives  entirely  satisfactory  results  in  both  manu- 
facture and  operation,  this  angle  being,  of  course,  measured  in 
a  plane  parallel  to  the  axis  of  the  worm. 

Throughout  what  follows  then,  a  value  of  60  degrees  will  be 
assumed  for  6. 

The  relation  between  the  axial  included  angle  and  the 
included  angle  measured  on  a  section  cut  normally  to  the  worm 
thread  is  such  that 

¥ 

6      tan  "2 

tan-=-  (19) 

2      cos  a 


Hence 


¥  6 

tan  —  =  tan  -  cos  a  (20) 


26 


WORM  GEARS 


and  the  milling  cutters  with  which  the  threads  of  the  worm 
are  milled  must  be  ground  to  this  angle,  W. 

Thirty  degrees  will  consequently  be  the  axial  pressure  angle 
recommended  for  general  practice. 

It  will  be  objected  that  the  above  practice  is  contradictory  to 
the  theory  of  reversibility — why  use  a  gliding  angle  so  much 
higher  than  the  theoretical  value?  The  answer  is  that  experi- 
ence shows  that  reversibility  is  not  affected  nor  is  the  efficiency, 
and  a  great  advantage  accrues  from  the  absence  of  any  under- 
cutting of  the  worm-wheel  teeth,  and  the  buttress  form  which 
is  given  them  greatly  increases  their  resistance  to  shearing  and 
diminishes  interference. 


Axial  Section  Through  Worm  Teeth 
FlG.   6. 

Fig.  6  illustrates  a  section  cut  through  the  worm  in  the  direc- 
tion of  the  axis  and  fixes  the  proportions  of  the  thread,  all  of 
which  are  based  upon  the  pitch  P,  which  is.  of  course  the  same 
as  the  circular  pitch  of  the  wheel.  The  points  of  the  teeth  are 
shown  with  sharp  corners,  but  in  practice,  when  finishing  the 
hardened  worm,  the  extreme  corners  should  be  very  slightly 
removed  by  grinding.  In  making  the  hob  for  cutting  the  worm 
wheel,  the  addendum  o  must  be  lengthened  as  shown  by  the 
dotted  lines  until  it  is  equal  to  KJ  by  which  means  a  working 
clearance  will  be  provided  at  the  bottom  of  the  teeth.  This 
will  give  to  the  wheel  teeth  a  dedendum  equal  to  that  of  the 


PRESSURE  ANGLE  AND  FORM  OF  THREAD   27 


worm  K,  and  the  height  of  the  teeth  in  contact  will  be  twice  the 
addendum  o  or  .6366  P. 

It  must  always  be  borne  in  mind  that  the  pressure  between 
the  worm  threads  and  the  wheel  teeth  is  transmitted  along  a 
line  normal  to  a  plane  drawn  mutually  tangent  to  the  teeth  of 
the  wheel  and  the  threads  of  the  worm  at  the  momentary  point 
of  contact.  When  the  worm  is  rotating,  a  number  of  these 
lines  of  pressure  envelop  the  worm  in  a  segment  of  a  cone,  whose 
apex  is  located  in  a  prolongation  of  the  worm  axis,  and  whose 
base  is  the  sector  of  the  circle  of  the  worm  pitch  line  contained 
within  the  angle  subtended  by  the  worm  wheel.  Hence  the 
thrust  of  the  worm  is  not  all  taken  up  on  the  thrust  bearing 
but  is  partly  carried  on  the  journal  bearings  in  which  the  worm 
is  mounted.  This  will  be  further  considered  when  dealing  with 
the  stresses  in  bearings. 

The  next  point  of  consideration  is  the  width  of  the  wheel; 
this  affects  two  things,  first  the  length  of  the  wheel  teeth,  and 
secondly  the  number  of  teeth  in  contact  with  the  worm. 

On  the  length  of  the  teeth,  to  a  great  extent,  depends  the 
strength  of  the  worm  wheel  considered  as  a  spur  wheel  and  this 
is  determined  as  follows : 


FIG.  7. 

The  tooth  pressures  will,  on  reflection,  be  seen  to  be  due  to 
one  of  two  causes. 

(a)  The  torque  of  the  motor  applied  at  the  worm  pitch  line. 

(b)  The  negative  tractive  effort  due  to  the  momentum  of  the 
car  when  brakes  are  applied  to  the  transmission. 

Dealing  with  (a),  in  Fig.  7,  the  developed  thread  surface  is 


28 


WORM  GEARS 


shown^by  the  line  AB,  the  developed  pitch  line  by  the  line  AC; 
T  is  the  torque  available  at  the  pitch  line.     Then  the  weight  W, 
which  can  be  lifted  by  T,  will  be  expressed  by  the  equation  : 
T 


Where  <f>  is  the  angle  of  friction,  the  reaction  between  the 
surfaces  at  p'  will  then  be  represented  by 

<22> 


This  reaction  is,  of  course,  the  tooth  pressure,  and  is  expressed 
by  the  equation: 

T 


tan 


P    = 

cos 
which  may  be  written 


(23) 


or 


P'  = 


[tan  (a 


P  = 


sn 


(24) 
(25) 


Messrs.  Brown  &  Sharpe  arrive  at  the  same  result  in  a  different 
manner.  Ignoring  friction,  they  resolve  the  acting  forces  as 
shown  in  Fig.  8. 

T 

(26) 


when 
hence 


cos 


T 


sin  90-7 


(27) 


PRESSURE  ANGLE  AND  FORM  OF  THREAD      29 

Neither  of  these  equations,  however,  takes  into  account  the 
pressure  angle;  for  the  more  accurate  expression  see  Equa- 
tion 43,  Chapter  VII. 

Many  forms  of  worm  teeth  are  possible.  Hitherto  we  have 
considered  only  straight-sided  teeth  but  the  worm  thread  may 
be  given  any  form,  for  example,  cycloidal  or  involute,  but  the 
difficulty  of  manufacturing  such  worms  as  these  is  almost  pro- 
hibitive and  a  straight  tooth  is  the  only  alternative.  A  curved 
tooth  worm  was  recently  made  and  tested  by  the  Brown  & 
Sharpe  Manufacturing  Company  in  the  experiments  which 
they  carried  out  early  in  1912,  but  beyond  being  an  extremely 
interesting  mechanical  production  and  working  quite  satis- 
factorily, the  worm  was  of  no  special  value.  Much  more  impor- 
tant, however,  is  the  hollow  or  Hindley  worm,  in  which  the  worm 
being  of  hour-glass  shape  is  made  to  embrace  an  arc  of  the  worm 
wheel.  That  perfectly  satisfactorily  working  gears  may  be 
designed  on  this  system  is  proved  by  the  uniform  success  ob- 
tained by  such  makers  as  Lanchester  and  Daimler.  There  is, 
however,  a  good  deal'of  misconception  as  to  the  relative  merits 
of  straight  and  curved  worms  which  it  is  the  object  of  the 
following  remarks  to  correct.  As  the  author  is  in  the  position 
of  having  employed  both  types,  he  will,  it  is  to  be  hoped,  be 
exonerated  from  any  prejudice  in  the  matter. 

In  the  first  place,  when  hobbing  the  wheel  for  an  ordinary 
straight  worm,  the  blank  is  entirely  grooved  out  from  a  solid 
mass  of  metal  by  the  hob  in  such  a  way  that  the  hob  completely 
generates  the  teeth  of  the  wheel  and  as  has  been  pointed  out, 
every  part  of  the  face  of  each  tooth  is  forced  to  touch  against 
some  part  of  the  worm  thread.  The  entire  surface  of  the  tooth 
is,  therefore,  subject  to  wear.  The  hob  is,  in  this  case,  at  its 
finishing  end,  an  exact  duplicate  in  its  profile  of  the  worm  itself. 

If  a  hob  be  made  of  the  curved  form  and  a  blank  be  cut  with 
it,  the  tooth  of  the  resulting  wheel  will  be  perfectly  straight 
and  the  wheel  will  be  flat  on  the  points  of  the  teeth.  Fig.  9 
is  a  photograph  of  such  a  wheel  cut  by  Brown  &  Sharpe  for 
the  experiments  already  referred  to.  It  is  seen  that  the  worm 
is  really  the  wheel,  and  the  wheel,  the  ordinary  straight  worm, 
the  relative  diameters  of  the  two  having  been  reversed  from  the 


30 


WORM  GEARS 


usual.  Such  a  gear  works  well  as  shown  by  the  experimental 
results.  So  far  as  the  author's  information  goes,  however,  it  is 
never  used  in  automobiles.  It  is,  nevertheless,  the  true  Hind- 
ley  worm,  and,  as  a  section  through  it  shows,  it  makes  contact 
at  all  points  across  the  teeth,  the  entire  tooth  being  subject  to 
wear  exactly  as  is  the  case  with  the  straight  gear,  which  in 
effect  it  is. 


FIG.  9. 

The  gear  first  employed  by  Lanchester  is  not,  however,  a 
truly  developed  gear  at  all  as  both  the  worm  and  the  wheel  are 
hollow  and  such  a  wheel  can  only  be  hobbed  with  a  very  short  hob 
and  driven  with  an  equally  short  worm.  It  has,  however,  this 
peculiarity  that  a  section  on  the  center  of  the  worm  shows  a 
perfect  contact  along  all  the  worm  teeth  at  once.  In  other 
words,  the  worm  applied  to  the  wheel  will  shut  out  daylight 
entirely  along  its  center  line  and  if  say  four  teeth  of  the  wheel 
are  subtended  by  the  worm,  all  will  be  in  contact  at  once.  In 


PRESSURE  ANGLE  AND  FORM  OF  THREAD      31 

Fig.  10,  which  represents  any  worm  gear,  let  the  lines  A,  B,  C,  D 
and  E  represent  planes  of  section.  Then  in  a  Lanchester  worm, 
a  section  of  plane  C  will  show  every  tooth  in  contact  over  its 
whole  depth.  Similar  sections  on  B  and  D  show  decreasing 
contact,  and  sections  on  A  and  E  show  no  contact  at  all.  If 
the  same  section  be  taken  on  a  straight  worm  gear,  it  will  be 
found  that  on  each  plane  there  is  one  tooth  in  contact  with  the 
worm  at  some  point  across  the  tooth.  Consequently  with  a 


D   E 


FIG.  10. 

very  little  care  on  the  part  of  the  designer,  the  number  of  teeth 
in  contact  can  be  made  exactly  the  same  in  the  case  of  the 
hollow  and  the  straight  worm.  This  point  must  be  insisted 
upon.  Here,  however,  there  comes  a  difference.  The  straight 
worm  perpetually  re-generates  the  surface  of  the  wheel  tooth 
maintaining  the  correct  involute  outline  to  the  end  of  its  life, 
and  in  the  other  type,  as  the  pressure  is  always  along  the 
central  plane,  it  follows  that  the  greatest  wear  takes  place  along 
this  line  and  the  tendency  is  to  finally  reduce  the  worm  wheel 
to  the  form  of  Fig.  11.  The  word  "  tendency  "  is  used  advisedly 
because  the  rate  of  wear  on  any  perfectly  made  worm  gear  is 
almost  inappreciable  so  that  the  difference  between  the  two  is 
rather  an  academic  one.  One  advantage  is  on  the  side  of  the 
Lanchester  worm;  since  the  sides  of  the  wheel  never  touch  the 


32 


WORM  GEARS 


worm,  they  can  be  cut  away  altogether  and  the  wheel  con- 
siderably reduced  in  width  and  weight  thereby.  Its  disadvan- 
tage is  that  the  worm  cannot  by  any  mechanical  device  be 
ground  on  any  of  its  finished  working  surfaces,  so  less  accurate 
finish  must  follow. 


FIG.  11. 

Another  argument  presents  itself  against  the  use  of  this  type : 
it  is  that  it  is  harder  to  assemble  and  adjust,  since  the  worm 
must  be  truly  located  on  the  wheel  in  three  planes,  while  the 
straight  one  requires  adjustment  in  only  two.  This  necessity 
for  exact  location  in  the  axial  direction  of  the  worm  demands 


PRESSURE  ANGLE  AND  FORM  OF  THREAD      33 

much  more  care  of  the  thrust  bearings  and  further  when  the 
worm  becomes  warm  under  use  expansion  must  take  place 
endways  and  the  contact  be  impaired.  The  fact  remains, 
however,  that  this  form  of  gear  is  entirely  satisfactory  if  per- 
fectly made  and  in  point  of  efficiency,  the  experiments  of 
the  Brown  &  Sharpe  Manufacturing  Company1  referred  to  in 
Chapter  X  show  that  the  efficiency  of  the  hollow  worm  ap- 
proaches that  of  the  straight  so  nearly  that  the  difference  may 
be  said  to  be  negligible  and  it  is  extremely  doubtful  if  a  suffi- 
cient number  of  examples  were  taken  whether  this  difference 
would  as  an  average  exist  at  all. 

1  The  Brown  &  Sharpe  experiments  referred  to  have  recently  (Nov.,  1912) 
been  confirmed  by  a  very  careful  test  on  a  Daimler  worm  driven  axle, 
carried  out  by  the  National  Physical  Laboratory,  London,  of  which  a 
report  has  been  published  by  the  Daimler  Motor  Car  Co.  of  Coventry, 
England. 


CHAPTER  VI 
STRENGTH  OF  WORM-WHEEL  TEETH 

The  behavior  of  a  worm  wheel,  considered  as  a  structure,  is 
exactly  analogous  to  that  of  a  spur  wheel;  that  is  to  say,  the 
stresses  upon  the  teeth  have  precisely  similar  effects  in  their 
tendency  to  shear  off  the  teeth,  and  therefore,  apart  from  the 
special  nature  of  worm  gear  and  the  effects  of  rubbing,  which 
have  to  be  provided  against  by  special  proportioning  of  the 
teeth,  it  becomes  necessary  to  design  them  from  this  point  of 
view  in  the  first  instance,  afterward  modifying  the  form  of  the 
teeth  if  necessary  to  suit  the  other  considerations  of  the  case. 


w 


FIG.  12. 


The  stresses  in  the  worm  wheel  caused  by  the  tangential 
pressure  between  it  and  the  worm  are  limited,  not  by  the  horse- 
power of  the  motor  but  by  the  maximum  tractive  resistance 
between  the  car  wheels  and  the  road,  thus 


(28) 

It 

The  tooth  of  the  worm  wheel  is  a  cantilever  on  which  the  load 
w  is  applied  at  a  point  near  the  center  of  the  tooth  on  the  pitch 
line  pi  (Fig.  12). 

34 


STRENGTH  OF  WORM-WHEEL  TEETH  35 

Let  Z  =  length  of  thread  of  worm  per    revolution    through 

360  degrees. 
/?  =  angle  of    worm  circumference  subtended    by  worm 

wheel. 
A  =  actual  length  of  tooth. 

P 

r  =  thickness  of  tooth  on  pitch  line  =  — 

Zi 

Then  1  =  -L  (29) 


The  length  of  the  cantilever  K  is  the  dedendum  of  the  tooth 
which  may  be  taken  as  .3683P. 

Let  a  =  permissible  stress  in  metal,  say  7000  Ib.  per  square 
inch. 

Then  the  safe  load  applied  to  the  cantilever  = 

/y 

W  =  —-  (where  Z  =  modulus  of  the  section)  (30) 

T2A 

and  Z==~Q  (31) 

If  x  =  safe  load  on  tooth  in  pounds 

*  =  6X360XT2Xg  (32) 

•3683P 
PV 

(33) 


•3683P 

This  simplifies  into 

x  =  2.2lpP  (34) 

This  formula  supplies  a  very  useful  check  on  the  design  of  the 
wheel  and  should  always  be  applied  before  finally  settling  the 
dimensions  of  the  gear.  Except  in  very  unusual  cases,  a  wheel 
properly  proportioned  as  regards  the  subtended  angle  /?  and  the 
number  of  teeth  in  contact,  will  be  strong  enough  in  any  case  as 
a  spur  wheel;  the  exceptions  exist  in  very  heavily  loaded  com- 
mercial vehicles  where  one  of  the  brakes  is  applied  to  some  part 
of  the  transmission  system,  whereby,  should  the  wheels  be 
locked  by  this  brake,  and  the  car  be  caused  to  skid,  the  tractive 


36  WORM  GEARS 

resistance  will  rise  to  a  high  figure  tending  to  shear  off  or  at 
least  to  bend  the  wheel  teeth,  while  at  the  same  time,  considered 
purely  as  a  worm  gear,  the  conditions  are  not  by  any  means 
.destructive  for  the  high  tooth  pressure  occurs  with  zero  rubbing 
velocity  and  consequently  there  are  no  heating  effects. 


CHAPTER  VII 


STRESSES  IN  WORM  GEARING 

In  a  subsequent  chapter,  the  stresses  in  the  gears  as  affected 
by  the  width  of  the  wheel  and  the  number  of  teeth  in  contact 
will  be  investigated,  but  before  considering  these,  it  will  be 
advisable  to  analyze  all  the  stresses  in  the  gears,  as  we  can  then 
determine  the  specific  tooth  pressure  between  the  working 
surfaces  and  this  value  will  be  needed  in  the  final  proportioning 
of  the  wheel. 


FIG.  13. 

In  the  example  already  considered,  the  contact  between  the 
worm  and  the  wheel  is  made  at  five  different  points  simultane- 
ously, but  owing  to  the  fact  that  there  are  five  threads  to  the 
worm,  any  diagrammatic  representation  of  the  same  is  much 
complicated;  let  us,  therefore,  consider  the  same  worm  divested 
of  four  of  its  threads  as  in  Fig.  13;  then  some  part  of  the  remain- 
ing thread  is  brought  in  contact  with  every  part  of  the  surface 
of  the  wheel  tooth  against  which  it  presses,  and  the  successive 
lines  of  pressure  commence  at  a  and  end  at  /.  The  surface  of 
the  worm  thread  forms  a  continuous  spiral  around  the  axis,  and, 
therefore,  each  one  of  these  lines  of  pressure  makes  successively 
the  same  angle  with  the  axis  of  the  worm.  In  Fig.  13,  these 

37 


38 


WORM  GEARS 


lines  of  pressure  are  shown  as  a  skeleton  diagram,  and  it  is  seen 
that  they  form  a  portion  of  the  surface  of  a  conical  spiral 
encircling  the  worm.  Taking  Fig.  13,  the  portion  of  this  surface 
will  be  seen  to  be  approximately  a  segment  of  a  cone,  and  seeing 
that  several  teeth  operate  at  one  time,  the  central  tooth  in 
contact  is  touching  the  worm  at  a  point  which  lies  in  a  line 
joining  the  centers  of  the  worm  and  worm  wheel  drawn  at  right 
angles  to  the  former;  the  other  teeth  approaching  to  and  receding 


FIG.  14. 

from  this  line,  touch  the  worm  threads  respectively  to  the 
right  and  left  of  this  point,  viewed  from  the  end  of  the  worm 
axis.  The  lines  of  pressure,  therefore,  for  every  point  of  con- 
tact range  themselves  in  such  directions  as  will  conform  more 
or  less  to  the  surface  of  a  cone,  and  the  resultant  line  of  pressure 
coincides  with  that  of  the  tooth  which  is  momentarily  in  the 
central  position.  Owing  to  the  fact  that  this  curved  surface 
is  not  a  true  cone  but  a  conical  spiral,  it  will  be  appreciated  that 
this  is  not  mathematically  accurate  but  sufficiently  so  for  our 


STRESSES  IN  WORM  GEARING  39 

purpose.  Taking  this  central  line  of  pressure  the  calculation 
of  the  direction  of  this  force  becomes  exceedingly  simple. 

Let  ab,  Fig.  14,  be  the  axis  of  the  worm  and  a  b  c  d  (7^)  a  per- 
pendicular plane  cutting  this  axis;  also  c  d  ef  (r2)  a  horizontal 
plane  normal  to  n1  and  touching  both  pitch  lines;  c  d  g  his  the 
cylinder  of  the  pitch  circle  of  the  worm,  and  I  p  m  that  of  the 
wheel;  p  is  the  point  of  contact  between  the  worm  thread  and 
the  central  tooth  in  engagement;  p  s  is  the  line  of  pressure  due 
to  the  angularity  of  the  lead  angle  a;  p  r  is  the  normal  pressure 

W 

angle  elevated  —  degrees  above  the  horizontal  plane.     Draw  r  o 

and  s  o,  each  perpendicular  to  the  line  dc;  raise  the  perpendiculars 
r  q  and  s  9,  and  through  the  point  q  where  they  cut,  draw  the 
line  p  q;  p  q  is  then  the  resultant  pressure  line;  p  q  does  not  lie  in 
the  plane  of  the  axis  a  b  and  therefore  p  q  and  a  b  will  never  meet 
although  not  parallel  to  each  other;  it  is  evident  that  the  force 
tending  to  separate  the  worm  and  the  wheel  passes  along  the 
line  p  q  and  is  the  reaction  of  the  tooth  pressure  at  this  point. 

r  o  s  q  is,  by  construction,  a  parallelogram;  hence,  p  s  q  is  a 
right  angle  and  if  p  s  be  drawn  to  such  a  scale  that  it  represents 
the  magnitude  of  the  tooth  pressure, 

(25)  Chapter  V 


The  length  of  the  resultant  p  q  can  be  calculated  as  follows 
Taking  first,  plane  n2  (Fig.  14) 
p  o  s  =  90  degrees 
/.    p  o    =  ps  cos  a 

and  so      =ps  sin  a 

In  plane  nlf 

W 

or  =  po  tan  —  (35) 

W 

and  pr  =  po  sec  —  (36) 

W 

.'.    or  =  ps  cos  a  tan  -  (37) 

2 

W 

and  pr  =  ps  cos  a  sec  —  (38) 


40  WORM  GEARS 

The  triangle  pqs  can  then  be  resolved  as  follows  : 
qs  =  or  and  psq  =  9Q  degrees 

(39) 


Substituting  (pq)2=(ps  cos  a  tan  -  )  +  (ps)2  (40) 


tan      -     +  (ps)2 


and  since  ps  is  by  construction  =  pf  =  — 


sin 

we  at  once  obtain  the  value  of  pq  the  resultant  pressure,  and 
since  pq  is  greater  than  ps,  the  value  of  pq  must  be  taken 
wherever  p'  is  in  question1  in  terms  of  the  torque  T. 

/T  cos  a  ¥\2       /     T   \2 

/^co    ?xt«\   +(^£.)  (42) 

sin  a  2  sm  « 


(43) 


and  pq  must  be  taken  as  the  correct  value  of  p'  wherever  the 
latter  is  required  to  express  the  actual  tooth  pressure  normal  to 
the  working  surface. 
Lastly,  qs  =  ps  tan  z 

tan  z  =  ^  (44) 

ps 

and  substituting 

P'  cos  «  tanT  f45) 

tan  z  =  —  — 

P 

W 

tan  2  =  cos  a  tan  -^-  (46) 

Zi 

Whence  z  is  easily  found  from  Equation  20,  Chapter  V. 

This  settles  the  direction  and  magnitude  of  the  resultant 
line  of  pressure  due  to  the  torque  of  the  motor. 

The  worm  and  worm  wheel  are  carried  in  ball  bearings  in 
which  the  stresses  can  be  found  in  the  following  manner.  The 
end  thrust  of  the  worm  in  the  direction  of  its  axis  is  po  (Fig.  15) 
and  is  equal  to  p'  cos  a. 

1  See  Equations  25,  26  and  27. 


STRESSES  IN  WORM  GEARING 


41 


but 
Therefore, 


sin  (a 
T  cos  a 
sin  C 


(47) 


P 


FIG.  15. 


It  has  already  been  shown  that  <p  is  negligibly  small. 
T  cos  a 


Therefore. 


po 


sn  a 


T  cot  a 


(48) 


po  is  the  tangential  pressure  on  the  worm  =  p 

.-.  p  =  T  cot  a  (49) 

Let  f=pq  =  ihe   separating   force,    qs  =  ro  =  ihe  force  acting 
radially  on  the  journal  bearings  of  the  worm. 


or  =  pf  cos  a  tan  — 


and  substituting  for  pf 


sin 


cos  a  tan  — 


(50) 
(51) 


42 
neglecting  <p 


WORM  GEARS 

W 
T  cos  OL  tan  — 

sin  a 

W 

.'.  or  =  rcot  a  tan  - 


(52) 
(53) 


Notwithstanding  that  this  vertical  force  is  the  actual  pres- 
sure tending  to  separate  the  worm  and  wheel  in  the  direction  of 
a  line  drawn  between  and  perpendicular  to  each  of  their  axes, 
it  is  not,  as  Mr.  Kalb  has  shown,  the  ultimate  stress  on  the 
journal  bearings  for  the  following  reason.  In  Fig.  17,  let  A  be 


FIG.  17. 

the  axis  of  the  worm,  T  is  the  moment  of  torque  at  the  pitch 
line;  it  is,  of  course,  balanced  by  the  reaction  R  which  is  the 
horizontal  stress  upon  the  journal  bearings  of  the  worm  =  T. 
We  have  just  seen,  however,  that  there  is  a  vertical  pressure  on 

W 
these  same  bearings  =  T  cot  a  tan—  =  Fin  the  figure.    There 

is,  therefore,  a  total  stress  on  the  journal  bearings  which  is 
equal  to  the  resultant  of  F  and  R}  viz.,  S;  from  the  construction 
the  figure  A  B  is  a  rectangle,  hence 


substituting  for  F2 


=      OT  cot  a  tan 


(54) 


(55) 


(56) 


STRESSES  IN  WORM  GEARING 


43 


S  being  in  this  case  the  resultant  separating  force.  It  is  evident 
that  this  force  acting  on  the  bearings  of  the  worm  has  its  exact 
counterpart  in  the  bearings  of  the  worm  wheel;  in  Fig.  18,  the 
two  side  components  of  the  forces  acting  on  the  wheel  are 
shown,  and  they  are  seen  to  consist  of  the  torque  T  and  the 
vertical  pressure  F  as  before;  the  resultant  force  S  produces  a 
diagonal  stress  on  the  worm  wheel  and  differential  case,  but  as 
the  vertical  and  horizontal  components  are  located  in  the  plane 


FIG.  18. 

of  the  axis  of  motion  in  the  case  of  the  wheel,  there  is  no  result- 
ant force  acting  on  the  bearings.  It  follows  from  this,  therefore, 
that  theoretically  the  journal  bearings  of  the  wheel  are  more 
lightly  stressed  than  those  of  the  worm,  but  in  practice  it  is  not 
possible  to  make  them  any  smaller,  even  if  it  were  desirable, 
since  the  arms  of  the  differential  case  are  invariably  larger  in 
diameter  than  the  worm  spindle  and  necessitate  a  larger  bearing 
to  support  them. 


44  WORM    GEARS 

With  regard  to  the  side  thrust  of  the  worm  wheel,  referring 
to  Fig.  14,  we  see  that  the  component  of  side  thrust  of  the  force 
pq  is  in  the  direction  so,  we  know  this  to  be  =  ps  sin  a,=p' 
sin  a 


p'=  ~     -- 
sin  (a+tp) 

T  sin  a 


but  ' 

f 

therefore  so=  —  — 

sin 

neglecting  <p  as  before 

so  =  T 

It  has  already  been  pointed  out  that  the  stresses  in  the  gears 
due  to  braking,  especially  in  the  case  of  a  heavy  commercial 
vehicle,  may  be  very  much  greater  than  those  due  to  the  motor 
torque.  Just  before  the  wheels  come  to  rest  by  skidding 
with  the  application  of  the  transmission  brake,  the  end  thrust 
on  the  worm  must  be  considered  to  be  that  due  to  the  tractive 
resistance  at  the  road  wheels,  multiplied  by  the  radius  of  the 
road  wheels,  and  divided  by  the  radius  of  the  worm  wheel.  As 
however,  at  the  instant  of  coming  to  rest,  that  is  when  the 
tractive  effort  is  so  low  as  to  be  negligible,  the  motion  of  the 
worm  is  very  slow,  the  ball  bearings  of  the  worm  thrust,  being 
scarcely  moving,  will  carry  a  very  great  load,  and  do  not  there- 
fore need  to  be  very  much  greater  than  is  necessary  for  satis- 
factory working  with  more  moderate  loads  at  high  speeds.  The 
conditions  under  which  the  vehicle  will  be  required  to  work  are, 
however,  so  various  that  the  risks  which  may  be  taken  in  this 
respect  must  be  a  matter  of  the  designer's  judgment  and 
experience,  and  no  empirical  rules  can  be  laid  down  for  them. 

It  need  hardly  be  pointed  out  that  the  torque  T  is  the  torque 
delivered  at  the  pitch  line  of  the  worm  when  the  lowest  trans- 
mission gear  is  in  use,  and  though  the  value  of  p'  v  does  not 
become  any  greater  (but  rather  less,  owing  to  the  work  absorbed 
in  tooth  and  bearing  friction  in  the  transmission  gears)  p'  is 
greater  than  the  torque  of  the  motor  itself,  by  the  amount  of  the 
gear  ratio  in  operation. 

We  are  now  in  a  position  to  tabulate  the  complete  stresses 
in  worm  gearing  for  any  angle  of  worm  gear,  and  their  magni- 
tude per  pound  of  torque  moment  for  all  usual  angles  of  lead  is 


STRESSES  IN  WORM  GEARING 


45 


given  in  the  following  table,  where  the  axial  pressure  angle  is 
assumed  to  be  30  degrees.     In  this  table  the  value  of  the  fric- 


Curves  showing  Stresses  per  Pound  of  Torque 
for  Various  Lead  Angles 
Axial  Pressure  Angle  =30° 


59 

58 

57 

56 

55 

54 

53  2 

52^ 

51-^- 


16       18      20       22      24       26      28       30       32      34       36       38      40      42      44      46 
Degrees  Lead  Angle  Cf 

FIG.  19. 

tion  angle  has  been  omitted  as  being  too  small  to  have  any 
practical  value. 


46 


WORM  GEARS 


Fig.  19  has  been  plotted  from  this  table,  and  from  it  all  the 
intermediate  values  of  different  angles  of  lead  may  be  found. 
This  diagram  will  be  found  of  great  value  in  designing  or 
analyzing  the  stresses  in  any  worm  gear. 

TABLE  III.— STRESSES  IN  WORM  GEARING 


Lead 
angle  a 

V 

Worm 
end 
thrust 

Resultant 
normal 
tooth 
pressure 

Separat- 
ing force 

Result- 
ant 
radial 
thrust 

Side 
thrust 

15° 

58°  18' 

3.732 

4.39 

2.16 

2.38 

1 

20° 

56°  50' 

2.747 

3.27 

1.485 

1.79 

1 

25° 

55°  14' 

2.144 

2.61 

1.122 

1.52 

1 

30° 

53°    8' 

1.732 

2.18 

0.866 

1.322 

1 

35° 

50°  38' 

1.428 

1.88 

0.676 

1.202 

1 

40° 

47°  44' 

1.191 

1.64 

0.527 

1.130 

1 

45° 

44°  26' 

1.000 

1.47 

0.405 

1.080 

1 

Lead  angle  =  a 


¥  is  obtained  from  tan  ^  =  tan  ~  cos  a 
Worm  end  thrust  =  T  cot  a 


(20)  Chapter  V 
(49) 


Vy\ 2    /  T  \ 2 
(T  cot  a  tan  2-  j  +  (  -.- j         (43) 

W 

Separating  force  =  T  cot  a  tan  —  (53) 

Radial  thrust 


/•  W\2 

=  \/(77cotatan  ^  )  +  T 
*  ^/ 


(56) 


CHAPTER  VIII 

THE  WIDTH  OF  THE  WORM  WHEEL 

When  an  automobile  is  being  driven  by  the  motor,  the  torque 
of  the  latter  is  applied  to  the  road  wheels  up  to  the  point  where 
they  will  commence  to  slip;  this  slip  will  prevent  the  increase 
of  horse-power  transmitted  beyond  a  certain  fixed  amount  (see 
equation  28,  Chapter  VI).  In  the  case  of  a  very  heavy  vehicle, 
the  tractive  effort  is  much  greater  than  the  motor  torque  and 
the  strength  of  the  wheel  teeth  x  (equation  34,  Chapter  VI) 
must  be  such  that  it  will  be  sufficient  to  withstand  the  tooth 
pressure  due  to  stresses  of  braking  calculated  from  equation  28. 

There  are  always  more  than  one  tooth  in  contact  at  one  time. 
Consequently, 

v' 
x  =  --,  and  we  can  write  (59) 

ilf  f 

997  /?p_  P 

£i*£i    l>  pJT  -  -~~7 

n 

P'  (60) 


~2.21P 

It  is  evident  from  this  that  ft  and  n'  vary  inversely,  and  $ 
cannot  be  diminished  without  correspondingly  increasing  nf ', 
and  since  n'  from  the  design  of  the  gear  is  unalterable,  it  follows 
that  if  ft  be  reduced,  the  equation  can  no  longer  be  satisfied. 

The  alteration  of  /?,  therefore,  has  a  very  important  effect  on 
the  behavior  of  the  gears,  since  the  specific  pressure  would  by 
its  decrease  be  augmented.  /?  is  thus  given  a  new  and  impor- 
tant significance,  for  in  addition  to  determining  the  strength  of 
each  individual  tooth,  it  also  governs  the  number  of  teeth  in 
engagement  at  one  time. 

47 


48  WORM  GEARS 

Mr.  John  Younger  has  supplied  the  following  means  of  obtain- 
ing the  value  of  ft 

The  length  of  the  worm  g  is  a  multiple  nf  of  the  circular 
pitch  P,  and  it  contains  in  its  length  a  certain  number  of 
threads  nf  of  which  it  is  most  important  to  take  the  fullest  pos- 
sible advantage.  In  order  to  ensure  this,  it  is  necessary  to  de- 
termine the  width  of  worm  surface,  or  in  other  words,  the  angle 
ft  which  will  provide  a  wheel  tooth  to  each  thread  of  the  worm. 
This  can  be  determined  graphically  in  the  following  manner : 

B  D 


IE 


FIG.  20. 


Let  A  B  in  Fig.  20  be  the  length  of  the  worm  =  g;  AC  is  the 
outside  circumference  of  the  worm.  From  the  point  A,  AD 
is  drawn  so  that  the  angle  DA  C  =a.  From  B,  the  line  BE 
is  drawn  perpendicular  to  AD.  The  distance  A  E  is  the  length 
of  the  arc  of  the  worm  which  must  be  subtended  by  the  worm 
wheel  in  order  that  some  part  of  every  thread  of  the  worm  in 
the  length  AB  will  touch  one  of  the  wheel  teeth;  from  this 
diagram,  the  following  equation  is  obtained  from  which  the 
value  of  /?  is  at  once  found  for  any  example. 


360  g  tan  a 


(62) 


THE  WIDTH  OF  THE  WORM  WHEEL 


49 


when  g  is  the  total  length  of  the  worm  =  n'P 
2Xo  =  2X.3183  P  =  .6366  P 
Hence  the  equation  becomes 

360  g  tan  a 


(63) 
(64) 


In  Fig.  21  the  cross-section  of  the  worm  wheel  is  shown.  It 
will  be  observed  that  the  "  width"  of  the  wheel  can  be  measured 
at  three  points,  viz.,  at  the  pitch  line,  at  the  root  of  the  teeth, 
and  at  the  outside,  this  last  being  somewhat  deceptive  as  it  has 
no  natural  connection  with  the  width  of  the  working  face. 


FIG.  21. 

Each  dimension  may  be  expressed  as  the  chord  of  the  angle 
/?  measured  at  different  radii.     The  radius  of  the  pitch  line 

being     -  that  of  the  root  of  the  teeth  ~  +  K. 
2  ,  2t 

Sufficient  thickness  of  metal  must  be  left  in  the  rim  to  prop- 
erly support  the  teeth  of  the  wheel  as  in  the  case  of  a  spur  gear. 

The  subject  of  worm  contact  has  been  very  fully  investigated 
by  Mr.  Robert  A.  Bruce  (Proc.  Inst.  Mech.  E.,  Jan.,  1906),  and 
the  reader  is  referred  to  this  for  a  very  complete  geometrical 
analysis  of  the  subject;  the  present  writer,  however,  takes 
exception  to  some  of  his  reasoning  which  is  somewhat  confusing 
by  his  method  of  illustrating  the  contact  by  the  use  of  vertical 
parallel  planes  cut  through  the  worm  teeth.  Such  planes  do 
not  follow  the  lines  of  pressure  between  the  contact  surfaces  and 
are  consequently  distorted  sections  through  what  is  admittedly 
a  very  difficult  section  to  correctly  illustrate  upon  paper. 


50  WORM  GEARS 

He  points  out,  however,  that  every  part  of  the  outline  of  the 
wheel  tooth  is  compelled  at  some  time  to  touch  some  part  of  the 
worm.  This  is,  of  course,  self-evident  when  it  is  remembered 
that  the  wheel  is  cut  with  a  hob  whose  outline  is  the  exact 
replica  of  the  worm. 

Mr.  Bruce  sums  up  his  conclusions  as  follows : 

"The  area  of  physical  contact  varies  as  the  pitch  diameter  of 
the  worm  multiplied  by  the  square  root  of  the  diameter  of  the 
wheel;  or,  if  the  effects  of  varying  the  angle  subtended  by  the 
pitch  line  of  the  worm  wheel  at  the  center  of  the  worm  be  con- 
sidered, it  may  be  said  that  the  effective  area  of  contact  varies 
as  the  continued  product  of  the  diameter  of  the  worm,  the 
tangent  of  half  the  angle  subtended  by  the  worm  wheel,  and  the 
square  root  of  the  diameter  of  the  worm  wheel.  At  any  instant 
the  end  pressure  is  shared  between  several  teeth,  and  it  is 
therefore  justifiable  to  expect  a  greater  power  of  sustaining 
loads  as  the  number  of  teeth  in  action  is  greater.  The  variation 
in  the  number  of  teeth  in  gear  is,  however,  much  more  apparent 
than  real.  Except  in  the  case  of  abnormally  small  worm  wheels, 
the  length  of  the  contact  paths  on  the  worm  wheel  side  of  the 
pitch  plane  is  unaffected  by  the  size  of  the  worm  wheel.  On 
the  other  side,  the  contact  lines  most  affected  are  those  which 
are  flattest,  or  which  most  nearly  coincide  with  the  pitch  line. 
The  actual  variation  in  the  number  of  teeth  in  gear  at  any  one 
time  is  found  on  careful  investigation  to  be  small  for  widely 
differing  sizes  of  worm  wheel.  So  that  in  comparison  with 
other  more  important  matters  it  may  be  neglected. 

"  By  keeping  the  ratio  of  the  height  to  the  thickness  of  the 
teeth  as  large  as  practicable,  the  greatest  possible  number  of 
teeth  are  enabled  to  operate  simultaneously,  but  at  the  same 
time,  in  order  to  avoid  interference,  the  teeth  should  be  pitched 
as  finely  as  is  compatible  with  strength  and  allowance  for  wear. 
It  should  be  noticed  that  in  respect  of  the  height  of  the  teeth 
the  dictum  given  above  is  in  direct  opposition  to  the  best  prac- 
tice with  spur  gearing,  where  entirely  different  conditions  are 
in  force. 

"The  effect  of  the  angle  of  the  worm  thread  remains  for  con- 
sideration. As  the  ratio  of  pitch  to  the  diameter  of  worm  be- 


THE  WIDTH  OF  THE  WORM  WHEEL  51 

comes  greater,  the  thrust  of  the  worm  is  borne  on  a  surface  of 
greater  inclination  and  the  actual  pressure  on  the  teeth  is 
increased  in  the  same  ratio  as  the  secant  of  the  angular  pitch. 
At  the  same  time  the  width  of  the  contact  line  across  the  face 
of  the  teeth  is  increased  in  the  same  ratio,  so  that  the  actual 
pressure  per  unit  of  width  remains  the  same.  It  is  not  necessary 
therefore  to  take  any  account  of  the  angle  of  the  helix  in  making 
estimates  of  the  'effective  contact  area'.  Under  precisely  similar 
conditions  as  to  temperature,  lubrication,  nature  of  the  surfaces 
in  contact  and  rubbing  velocities,  it  might  reasonably  be 
anticipated  that  the  end  thrust  would  be  proportional  to  the 
effective  area,  and  neglecting  comparatively  unimportant 
factors  the  relation  may  be  expressed  as  follows: 


p  =  K^Ddtsm^  (65) 

so  p  =  some  factor  depending  on  conditions  X  effective  breadth 

X  effective  width  across  the  face  of  the  worm  teeth, 
where  p   =safe  end- pressure  in  pounds. 

d    =  diameter  of  worm  at  pitch  line. 

D  =  diameter  of  worm  wheel  pitch  line. 

/?  =  angle  subtended  by  the  worm  wheel  at  the  axis  of 
the  worm. 

K  =  a  factor  depending  upon  rubbing  velocity,  nature  of 
the  surfaces,  temperature  and  nature  of  the  lubri- 
cant, etc. 

"  Experience  has  shown  that  this  relationship  is  far  from 
simple  in  practice,  because  of  the  great  difference  in  the  factor 
K  imposed  by  variable  conditions.  Broadly  speaking,  what 
happens  in  the  case  of  a  worm  wheel  drive  is  very  much  what 
happens  in  the  case  of  a  loaded  journal.  The  action  commences 
under  certain  conditions  as  to  speed,  temperature  and  so  forth, 
and  as  it  proceeds  heat  is  generated  owing  to  frictional  resistance, 
the  amount  depending  upon  the  load,  the  lubricant  and  the 
efficiency  of  the  gear.  The  temperature  of  the  system  rises 
until  the  heat  generated  by  friction  balances  the  heat  lost  by 
radiation  and  convection,  when  a  stable  set  of  conditions  is 


52  WORM  GEARS 

established.  But  while  the  temperature  is  rising,  the  lubricant 
is  losing  its  viscosity,  and,  though  this  tends  to  diminish  the 
friction  and  consequently  the  generation  of  heat,  it  nevertheless 
diminishes  the  power  of  sustaining  a  heavy  load.  A  worm  will, 
therefore,  be  successful  if  the  viscosity  of  the  lubricant  does  not 
diminish  to  such  an  extent  that  its  load-sustaining  properties 
are  neutralized.  If  the  surfaces  be  allowed  to  come  into  grind- 
ing contact,  further  heating  takes  place,  and  the  lubricant 
becomes  still  less  viscous  and  therefore  incapable  of  bearing  a 
load,  and  seizing  will  take  place  quickly." 

Mr.  Bruce,  in  his  paper  already  referred  to,  gives  the  follow- 
ing formula  for  calculating  the  width  of  contact  area  from 
which  he  determines  the  tooth  pressure.  According  to  this, 


(66) 


when  rl  and  rz  are  the  respective  radii  of  the  surfaces  in  contact. 
As,  however,  owing  to  exigencies  of  interference  when  being 
hobbed,  it  is  practically  impossible  to  determine  the  radius  of 
the  wheel-teeth  surfaces,  this  formula  does  not  seem  to  have 
much  more  than  a  theoretical  value.  Moreover,  owing  to  the 
elasticity  of  the  metals  employed,  and  variations  in  the  quality 
of  the  lubricant,  -no  reasonable  figure  can  be  assigned  to  K  in 
the  equation.  All  that  we  can  do,  therefore,  is  to  establish 
from  experience  an  empirical  value  for  the  number  of  teeth  in 
contact  in  its  relation  to  actual  performance,  and  deduce  some 
rule  from  this. 

The  author  carried  out  the  following  experiment  to  ascertain 
the  effect  of  varying  /?  in  a  specific  instance. 

An  axle  was  taken  fitted  with  a  standard  worm  gear  similar 
to  many  hundreds  running  in  heavy  commercial  vehicles  of  the 
author's  design  and  giving  eminently  satisfactory  results.  The 
gear  was  driven  from  a  50-H.  P.  electric  motor  running  at  a 
constant  speed,  and  an  hydraulic  dynamometer  was  attached 
to  the  other  side  of  the  gear,  and  adjusted  to  furnish  a  constant 
load  (Fig.  21  A).  The  dimensions  of  the  worm  gear  were  as 
follows  : 


THE  WIDTH  OF  THE  WORM  WHEEL  53 


54  WORM  GEARS 

Worm:  Number  of  threads 5 

Pitch  diameter 3.25  in. 

Pitch  circumference 10.21  in. 

Length  of  worm 5  in. 

Lead 5.  9375  in. 

Lead  angle 30°  11' 

Thread  angle 60° 

Length  of  thread  per  rev 11.812  in. 

Rubbing  speed 18.5  /.  p.  s. 

Worm  Wheel:  No.  of  teeth 39 

Pitch  diameter 14.75  in. 

Pitch 1.875  in. 

Subtended  angle  /? (see  table) . 

The  torque  at  the  worm  pitch  line  was  kept  constant  at 
1990  Ib.  giving  a  normal  tooth  pressure  of  4378  Ib.     The  rubbing 


speed  was  also  nearly  constant  at  18.5  ft.  per  second.  Conse- 
quently the  heat  generated  in  friction  (from  equation  68,  Chap- 
ter IX)  was  12.6  B.T.U.  per  minute  and  (from  equation  73, 
Chapter  IX)  the  area  of  rear  axle  surface  should  be  8.6  sq.  ft. 


THE  WIDTH  OF  THE  WORM  WHEEL 


55 


The  actual  surface  was  approximately  11  sq.  ft.  Fig.  22  shows 
a  section  of  the  worm  wheel  of  which  the  original  value  of  /? 
was  107°  45';  under  these  conditions,  the  number  of  teeth  in 
contact  was  5  (equation  12,  Chapter  IV).  The  wheel  was  subse- 
quently cut  down  in  width  as  shown  in  the  figure. 

In  Fig.  22,  the  gears  are  shown  diametrically  in  section  and 
plan,  the  lines  AB,  CD,  indicate  the  original  width  of  the  gears 
with  /?  =  107°  45'.  There  are  then  five  points  of  contact — a,  b, 
c,  d,  and  e. 

/5  was  then  reduced  to  72°  as  shown  by  lines  EF  and  GH. 
The  points  of  contact  are  thus  reduced  to  three — b,  c,  and  d. 

/?  reduced  again  to  64°,  still  left  three  points  of  contact,  but 
further  reduction  to  55°  30'  as  shown  by  lines  //,  KL  left  only 
two  teeth  in  contact  at  c. 

The  following  table  shows  the  results  obtained: 

TABLE  IV 


No.  1 

No.  2 

No.  3 

No.  4 

Subtended  angle  /?  

107°  45' 

72° 

64° 

55°  30' 

Revo   of  worm 

2  000211 

775  008 

1  590  000 

Failed 

Rubbing  velocity  v.  f  .p.s  
Tooth  pressure  p'  
Area  at  base  of  tooth 

18.5 
4378 
3  286  in 

18.25 
4378 
2  262  in 

18.34 
4378 
2  1  in 

18.34 
4378 
1  82  in 

Width  of  wheel  

3.25  in. 

2.  3437  in. 

2.0937  in. 

1  8437  in 

Temperature  attained  t'  
Temperature  of  air  t 

163°F. 

75° 

412°  F. 

76° 

412°  F. 

74° 

500°  F. 

72° 

t'-t  

88° 

336° 

338° 

428° 

No.  of  teeth  in  contact  
Pressure  per  tooth 

5 

875 

3 
1459 

3 
1459 

2 

2189 

Estimated  pressure  per  square 
inch. 

5840 

9730 

9730 

14,580 

The  failure  of  the  gears  in  Experiment  No.  4  took  place  from 
partial  seizure.  In  the  first  three  experiments,  the  duration 
was  sufficient  to  arrive  at  constant  conditions  and  a  maximum 
temperature  rise,  but  about  twenty  minutes  running  under 
conditions  No.  4  showed  the  temperature  rising  so  rapidly  that 
operations  were  suspended  and  the  gears  examined.  It  was 
plainly  evident  that  failure  was  due  to  actual  metallic  contact 
5 


56  WORM  GEARS 

between  the  surfaces,  and  this  fact  taken  in  conjunction  with 
the  last  lines  of  the  table  shows  an  unmistakable  connection 
between  specific  tooth  pressure  and  temperature.  The  condi- 
tions in  No.  1  test  being  alone  normal  when  the  individual  tooth 
pressure  was  875  Ib. 

To  determine  the  area  in  contact  on  each  point  is  impossible 
with  any  accuracy  but  in  the  author's  opinion,  in  the  gears  in 
question,  this  amounted  to  .15  to  .2  sq.  in — probably  the 
former.  Assuming  this  figure,  we  should  have  in  the  first 
experiment  5840  Ib.  per  square  inch,  and  since  experiment  shows 


FIG.  23. 

that  the  dry  metal  will  carry  this  pressure  within  its  elastic 
limit,  there  is  every  reason  to  believe  it  was  not  much  exceeded. 
Plotting  the  pounds  per  tooth  in  the  form  of  a  curve  (Fig.  24), 
we  can  obtain  an  approximation  to  the  intermediate  points, 
and  we  find  the  curve  to  be  distinctly  hyperbolic  in  character  as 
might  be  expected,  having  its  o  origin  with  an  infinite  number 
of  teeth  and  rising  to  infinity  with  o  teeth. 

All  that  can  be  said  positively  about  the  contact  is  that 
undoubtedly  the  surfaces  are  separated  by  a  film  of  oil  (Fig.  23), 
and  between  the  two  convex  surfaces  the  existence  of  this  oil 
cushion  can  be  relied  upon  provided  an  individual  tooth  pressure 
of  say  900  Ib.  is  not  exceeded  at  a  velocity  of  18.5  ft.  per  second. 
Multiplying  these  values  together,  we  at  once  obtain  the  neces- 
sary constant, 

2/0  =  16,650  (67) 

and  it  may  be  safely  said  that  17,000  must  never  be  exceeded  for 
continuous  running  if  overheating  is  to  be  avoided. 

The  fact  will  not  be  lost  sight  of  that  in  all  four  of  the  above 
mentioned  experiments,  the  quantity  of  heat  generated  was 


THE  WIDTH  OF  THE  WORM  WHEEL 


57 


approximately  constant,  and  of  course,  the  surface  of  radiation 
remained  the  same;  the  convection  and  radiation  of  heat  must, 
therefore,  have  been  at  a  constant  rate  also,  and  the  rise  in 
temperature  was  due  to  an  alteration  in  the  coefficient  of  fric- 


Lbs.  Pressure  per  Tooth  in  Momentary  Contact 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

jv 

\ 

\ 

\ 

234 
No.  of  Teeth  in  Contact 

FIG.  24. 

tion,  caused  by  the,  at  first,  partial,  and  subsequent  total, 
breaking  down  of  the  oil  film  with  the  increased  tooth  pressure. 
Too  much  stress  cannot,  therefore,  be  laid  upon  the  importance 
of  securing  a  proper  lubricant.  In  these  trials,  the  oil  used  was 
what  is  commercially  known  as  "600  W  cylinder  oil/'  this 
being  a  mixture  of  heavy  mineral  oil  of  high  viscosity  with 
some  animal  oil  and  possessing  the  following  properties : 

BeaumS  gravity  25 . 9°  at  60°  F. 
Specific  gravity  .  8980  at  60°  F. 
Viscosity  154.4  at  210°  F. 

The  viscosity  reading  is  with  a  Tagliabue  viscosimeter. 
It  was  found  that  a  sample  of  this  oil  could  be  heated  to 
700°  F.  in  an  open  cup  without  igniting. 


CHAPTER  IX 

THE  TEMPERATURE  COEFFICIENT 

In  any  worm  gear,  there  must  always  be  a  definite  amount  of 
heat  generated  since  it  cannot  operate  without  some  friction; 
which  is  converted  into  heat. 

Let  H  be  the  quantity  of  heat  in  B.T.U.  per  minute. 


Then  tt-~  (68) 


This  quantity  of  heat  is  added  every  minute  to  the  complete 
axle  system  which  will  raise  its  temperature  to  a  definite  extent; 
and  if  h  =  mean  specific  heat  of  the  whole  axle  system  and  m 
its  mass  in  pounds,  its  capacity  for  absorbing  this  heat  will  be 
mh.  Then  if  no  radiation  takes  place  from  the  axle,  its  tem- 
perature will  increase  at  the  rate  of 

TT 

—T  degrees  F.  per  minute  (69) 

It  is  evident  that  the  heat  generated  has  ultimately  to  be 
radiated  from  the  axle  at  the  same  rate  per  minute.  Let  q  = 
this  quantity  of  heat  in  B.T.U. 

It  is  desirable  that  the  oil  should  not  be  permitted  to  attain  a 
temperature  of  over  180°  F.  because  its  viscosity  and  lubricating 
properties  are  considerably  reduced  at  higher  temperatures, 
and  if  we  assume  the  worst  conditions  and  provide  for  an  atmos- 
pheric temperature  of  100°  F.,  it  follows  that  the  difference 
between  the  temperature  of  the  worm  and  the  atmosphere  will 
be  80°  F. 

Let  If  and  t  respectively  be  the  temperatures  of  the  gear  and 
the  atmosphere.  Then 

tf-t  =  SQ°  (68) 

58 


THE  TEMPERATURE  COEFFICIENT  59 

Rankine's  formula  for  radiation  is 


where  a=  surface  in  square  feet 

802      106.6 


Connecting  equations  (68)  and  (71)  we  get 

3=ff 
or 

106.6 


778 
.0771 


, 


106.6  1383     691,500 

(7o) 


.0771  p'fiv         p'nv         p'v 

This  figure  is  necessarily  only  approximate  because  the  rate 
of  radiation  varies  with  so  many  factors.  For  example,  the 
condition  of  cleanliness  or  otherwise  of  the  axle  casing  and  its 
speed  through  the  air,  atmospheric  conditions,  and  so  forth. 
Moreover,  heat  is  conducted  away  from  the  gearing  through  the 
axles  and  springs  and  to  some  extent  through  the  propeller 
shaft.  It  will  be  found  in  practice  that  the  surface  of  radiation 
determined  by  the  formula  is  quite  easy  to  obtain,  but  if  from 
any  cause  it  is  too  small,  overheating  will  be  only  a  question  of 
how  often  the  vehicle  has  to  run  at  maximum  power  on  low 
gear,  when  conditions  of  tooth  pressure  will  be  highest.  Even 
in  this  direction,  some  chances  may  legitimately  be  taken,  since 
continuous  running  on  low  gear  very  rarely  occurs,  and  even 
on  long  hills  there  are  periods  where  the  maximum  horse-power 
is  not  exerted,  which  will  consequently  tend  to  a  heat  balance 
being  maintained. 

It  must  be  pointed  out  that  the  oil  used  for  lubrication  is  but 
a  poor  conductor  of  heat  so  that  the  difference  in  temperature 
between  the  gears  and  the  atmosphere  is  generally  greater  than 
is  rendered  apparent  by  a  thermometer  placed  in  the  oil,  but 


60 


WORM  GEARS 


the  gears  may  be  run  for  short  periods  at  very  much  higher 
temperatures  than  those  allowed  for. 

In  formula  (68)  since  /*  is  a  constant  quantity,  viz  .  .002,  it 
follows  that 

(74) 


when  K  is  a  constant  =  .000154. 


Tooth  Pressures-  Lbs.  p  ' 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

I 

'I 

5C 

60 

0 

\ 

\ 

\ 

\ 

s 

X. 

v- 

^ 

X. 

^ 

X 

v 

X 

^ 

**• 

^ 

~- 

-^ 



—  ^ 

10 


15 


25  30 

Velocities  -  Ft.  per  Sec. 

FIG.  25. 


35 


40 


45 


and  consequently  H  varies  as  p'V,  and  if  a  constant  value  is 
demanded  for  H,  it  follows  that  p'V  is  a  constant. 

In  Fig.  25,  an  hypothetical  case,  the  rubbing  velocities  are 
represented  as  abscissae  and  the  tooth  pressure  in  pounds  as 
ordinates,  ^'^  =  50,600  a  constant  and  the  curve  is  in  conse- 
quence an  hyperbola. 

In  the  Bach  &  Roser  experiments,  it  was  found  that  at 
constant  temperatures,  a  very  close  approximation  to  this 
hyperbola  was  obtained.  Thus,  the  important  deduction  from 
this  is,  that  given  the  maximum  pressure  and  speed  of  any  gear 
at  which  a  constant  temperature  may  be  maintained,  the  other 
pressures  and  speeds  can  always  be  calculated,  lower  speeds 
permitting  higher  pressures  and  conversely. 


THE  TEMPERATURE  COEFFICIENT 


61 


It  is  profitable  to  investigate  the  effect  on  the  quantity  of 
heat  generated  by  varying  the  pressure  and  the  velocity. 

60 


Case  1. — Assume  p'  constant  at  3500  Ib.  then 
3500  X. 002X60 


778 

TABLE  V 

v  f.p.s. 

B.T.U.  per  minute 

5 
10 

2.70 
5.4 

15 

8.1 

20 

10.8 

25 

13.5 

30 

16.2 

35 

18.9 

40 

21.6 

45 

24.3 

Here  the  difference  =  2. 7  B.T.U.  per  increment  of  5  f.p.s. 
Case  2. — Assume  v  constant  at  45  ft.  per  second 

,.002X45X60 


P          778 

TABLE  VI 

Pf 

B.T.U.  per  minute 

1500 

10.41 

1750 

12.14 

2000 

13.88 

2250 

15.62 

2500 

17.35 

2750 

19.09 

3000 

20.81 

3250 

22.55 

3500 

24.30 

3750 

26.05 

4000 

27.75 

02 


WORM  GEARS 


Here  the  difference  =  7.4  B.T.U.  per  increment  of  250  Ib.  Fig. 
26  shows  these  increments  of  H  plotted  against  pressures  and 
velocities  respectively,  and  the  results  show  that  for  a  com- 


^.^x 

Normal  Tooth  Pressure 

^^xi: 

^ 

|xXx 

^ 

•-<& 

^  

•^^x 

r^^ 

^ 

^ 

i-O*^ 

X 

•<^2n 

^x 

^* 

•"^""^ 

<f&\ 

&s^ 

^  ' 

v^ 

^^ 

.s' 

^"^ 

S 

10        12       14        10        18 
B.T.U.  per  Minute 

FIG.  26. 


20        22        24        26       28 


paratively  small  increase  in  the  normal  tooth  pressure,  the 
heating  effect  is  increased  at  a  greater  rate  than  with  relatively 
larger  increments  of  the  velocity. 


CHAPTER  X 

EFFICIENCY  OF  WORM  GEARING 

The  efficiency  of  worm  gearing  has  been  discussed  by  several 
writers  and  it  will  be  dealt  with  next,  because  closely  bound  up 
with  it  is  the  coefficient  of  friction  between  the  worm  and 
wheel,  and  the  determination  of  the  latter  is  a  necessary 
preliminary. 

The  most  complete  investigation  of  mechanical  friction  that 
the  author  is  familiar  with,  is  that  conducted  by  Beauchamp 
Tower,  and  published  by  him  in  various  numbers  of  the  Proceed- 
ings of  the  Institution  of  Mechanical  Engineers.  Taking  the 
materials  most  nearly  corresponding  to  those  employed  for 
worm  gears,  viz.,  steel  rubbing  against  bronze  in  a  bath  of 
mineral  oil,  we  find  the  lowest  value  he  obtained  for  //  was  .0008. 
No  doubt  these  conditions  were  ideal,  but  suppose  we  take  a 
higher  value,  .002,  as  an  assumption,  and  then  see  how  nearly 
this  can  be  approached  and  maintained  in  practice;  it  will 
presently  be  shown  that  we  are  fully  justified  in  so  doing,  see- 
ing that  the  conditions  favorable  to  a  low  coefficient  are  all 
present  in  a  worm  gear.  These  are 

1.  Brief  period  of  contact. 

2.  Intermittent  pressure. 

3.  Certainty  of  oil  films  reaching  every  part  of  the  pressure 
surface. 

4.  No  metallic  contact. 

The  well-known  experiments  of  Bach  &  Roser  at  the  Royal 
Technical  High  School,  Stuttgart,  throw  some  light  on  the 
behavior  of  worm  gears,  but  unfortunately  they  were  carried  out 
upon  gears  which  were  by  no  means  as  accurately  made  as  is 
the  case  in  the  present  day  practice;  but  while  their  results  are 
of  little  specific  value,  they  undoubtedly  illustrate  the  laws 
which  govern  the  whole  performance  of  worm  gears. 

63 


64  WORM  GEARS 

The  worm  gear  upon  which  the  experiments  were  made 
appears  to  have  been  of  the  following  proportions: 

Worm :  Number  of  threads 3 

Pitch  diameter 3  in. 

Lead .  3  in. 

Lead  angle 17°  40' 

Wheel:  Number  of  teeth 30 

Pitch  diameter 9 . 54  in. 

Circumferential  pitch 1  in. 

The  tooth  pressures  were  varied  from  190  to  2660  lb.,  and  tests 
were  made  at  six  different  speeds  as  follows: 

v  =28.30  f.p.s. 
19.34f.p.s. 
9. 80  f.p.s. 
4.62  f.p.s. 
2. 56  f.p.s. 
0.85  f.p.s. 

It  will  be  observed  that  while  the  tooth  pressures  are  in  some 
accordance  with  automobile  practice,  the  velocities  are  lower 
than  is  ordinarily  the  case. 

Fig.  27  shows  the  tooth  pressures  and  coefficients  of  friction 
for  different  velocities  obtained  in  these  experiments.  It  is  at 
once  apparent  that  for  velocities  between  2.5  and  20  ft.  per 
second,  the  coefficient  is  but  little  affected  by  the  speed  but 
varies  curiously  with  the  pressures  on  the  teeth,  attaining  in 
all  cases  a  minimum  at  or  near  1000  Ib.  tangential  pressure. 
Unfortunately  these  experiments  do  not  state,  or  give  sufficient 
data  to  calculate,  the  specific  tooth  pressure  per  square  inch, 
which  would  have  been  very  valuable.  As  has,  however,  been 
pointed  out  by  Mr.  Robert  A.  Bruce,  the  deductions  from  these 
experiments  are  only  useful  when  the  exact  conditions  of  the 
original  experiments  are  reproduced,  and  he  further  points  out 
that  with  superior  working  surfaces,  e.g.,  hardened  and  ground 
worms,  and  efficient  cooling  arrangements  higher  pressures 
might  be  realized. 

With  a  view  to  proving  the  comparability  of  worm  gears 
and  th,e  journals  investigated  by  Beauchamp  Tower,  the  author 
carried  out  the  following  experiment. 

A  completely  assembled  rear  axle  belonging  to  a  5-ton  com- 


EFFICIENCY  OF  WORM  GEARING 


65 


mercial  vehicle  of  the  author's  design  was  taken  and  mounted 
upon  a  stand  (Fig.  28),  using  one  of  the  road  wheels  as  a  drum 
round  which  a  weighted  cord  could  be  wound.  A  smaller 
drum  was  secured  to  the  worm  spindle  with  another  weighted 
cord  wound  upon  this.  The  relative  diameters  of  the  drum  on 
the  axle  and  the  drum  on  the  worm  spindle  were  such  that  their 
ratio  was  the  same  as  that  of  the  worm  gear  reduction,  viz.,  7.8 
to  1.  Equal  weights  being  hung  upon  the  cords,  equilibrium 


Coefficients  of  Friction 
P  P  P  P  P  P  P 

_g  S  8  g  g  g  S 

-  U 

\\ 

JBj 

t-"'    ^ 

= 

19.34  f  .p 

.s. 

/ 

E 

\ 

0. 

=  21 

.3 

t 

\ 

8 

/ 

vy 

e 

\\ 

£ 

=  ( 

).8£ 

/, 

'// 

\\ 

\ 

/*> 

y7 

S 

// 

'/ 

\ 

\\ 

v 

-  — 

•^— 

^^ 

— 

/ 

'  / 

: 

A 

A 

\ 

f 

A 

A 

V- 

.8 

/ 

/ 

1 

'\ 
k 

^/ 

/ 

< 

\/ 

t'= 

=  4. 

62 

( 

^ 

t 

X 

/ 

' 

\ 

\ 

. 

ft 

/ 

/ 

X 

/ 

\ 

S 

^ 

F 

> 

',/. 

/ 

A 

^. 

^ 

^^ 

/ 

X 

/ 

V 

.  

. 

^« 

^ 

0 

»< 

.56 

BAG 
Relation  bet\ 
and  Coeff 

H  &  ROSER 
veen.  Pressure,  Velocity 
icient    of  Friction 

- 

)                     500                   1000                  1500                  2000                2500                300C 

Tangential  Tooth  Pressures  -  Lbs. 

FIG.  27. 


was  established.  The  mechanism  being  set  in  motion  by  hand, 
weights  were  added  to  the  cord  on  the  worm  drum  until  a  con- 
stant speed  of  rotation  could  be  maintained. 

On  substituting  heavier  weights,  it  was  found  that  a  greater 
weight  was  needed  to  keep  the  wheels  in  motion,  and,  several 
observations  being  thus  made,  a  number  of  values  were  obtained 
for  the  weight  necessary  to  maintain  uniform  motion  with 
different  degrees  of  tooth  pressure.  Plotting  these  values  at 
ordinates,  upon  tooth  pressures  as  abscissa?,  it  was  found  that 


66 


WORM  GEARS 


EFFICIENCY  OF  WORM  GEARING 


67 


these  fell  in  a  straight  line  as  shown  by  the  line  A  (Fig.  29) 
The  observed  points  being 

TABLE  VII 


Pressures  x 

Load  to  overcome  friction  y 

21b. 

3.75 

4  Ib. 

4.75 

81b. 

6.75 

It  is  evident  that  the  loads  y  are  the  product  of  x  X  /* 


.2  5 


«w  4 

s 


Curves  showing  Values  of  H  from  Observed  Friction  Readings 
from  Observed  Points.       V  =  a?/*  &  2y-X=5.5   .'.  2XfA-x=5.5 
fj.  approaches  \  2  as  a  Limit  as  X  approaches  oo 
ju,           ««          co  «<  ««     ««      „   ««          «           Zero. 

| 

1 

j 

/ 

->j 

|] 

rf 
J 

/ 

-1 

h 

-A 

1 

/ 

f 

/ 

, 

/ 

1 

\ 

5.5  +£ 

2^ 

\ 

\ 

^^ 

—  -  — 

*—  — 

—  —  . 

—  — 

_.._  ' 

= 

-==— 

•••jij- 



2    4     6     8    10 


20  30 

Pressure  in  #X 

FIG.  29. 


40 


The  equation  to  a  straight  line  is 


50 


Fig.  29  shows  the  curve  to  this  equation  giving  values  of 
for  various  pressures  x,  from  which  we  see  that 


WORM  GEAR'S 


H  approaches  .  5  as  a  limit  as  x  approaches  oo 
/(  approaches  cx>  as  a  limit  as  x  approaches  o. 

Turning  again  to  Beauchamp  Tower's  experiments,  the 
curves  in  Fig.  30  show  the  relationship  between  bearing  pres- 
sures and  /i  for  three  different  rubbing  velocities,  from  which  it 
is  evident  that  the  characteristic  of  the  curve  obtained  from  the 
entire  axle  is  identical  with  these  and  consequently  the  rules 
deduced  from  Tower's  experiments  by  Unwin  for  the  variations 
of  fjL  with  pressures  and  speeds  will  apply  equally  to  worm 
gears.  This  is  highly  important  in  all  that  follows. 

Beauchamp  Tower's  observed  figures  for  mineral  oil  bath  lubrication. 
Frictional  resistance  R =juP  =  PC  A/- 
Mean value  of  C  for  mineral  oil  .27 
0.012 1 


0.011 
0.010 

ao.009 

33  0.008 

.— 

£  0.007 

1 0.006 

|  0.005 

§0.004 

0.003 

0.002 

0.001 

0.000 


C\ 


(a)  0.426  0.26 
(i)  0.56  0.27 
(C)  0.733  0.28 


(c)    /*  approaches  0.0023  as  Limit 
(6)    -          "  0.0017  «       « 

(a)   »        «•         0.0014  » 

As  #  approaches  co 


400 


100  200  300 

Bearing  Pressures,  Lbs.  a  " 

X 

FIG.  30. 

It  will  at  once  be  objected  that  the  values  of  /*  obtained  by 
experiment  are  much  greater  than  .002;  the  reason  for  this  is 
very  plain.  In  the  axle,  as  tested,  there  was  the  friction  of  the 
ball  bearings  supporting  the  worm  and  the  worm  wheel,  and 
also  the  friction  of  the  roller  bearings  on  which  the  heavy  road 
wheels  were  mounted  on  the  axle  tubes;  this  friction  would 
undoubtedly  be  a  constant,  following  Unwin's  rule,  and  the 
total  resistance  to  be  overcome  by  the  falling  weight  is  thus 


EFFICIENCY  OF  WORM  GEARING 


69 


We  are  not  much  concerned  with  this  constant  since  it 
includes  all  losses  in  addition  to  the  worm  losses  proper,  but 
the  point  has  now  been  proved  that  the  sliding  friction  of  worm 
gear  is  substantially  the  same  as  the  sliding  friction  of  a 
journal  in  its  bearing,  and  we  are,  therefore,  justified  in  assum- 
ing the  same  specific  value  for  ^  as  in  the  case  of  a  journal. 
We  will,  therefore,  retain  the  value  of 


=  .002  = 


(75) 


and  presently  show  that  this  is  correct. 

The  losses  in  a  worm  gear  may  be  demonstrated  graphically 
in  the  following  manner :  We  may  first  consider  the  worm  as  a 
continuous  inclined  plane  and  the  teeth  of  the  worm  wheel  as 
the  weight  which  is  to  be  raised  up  this  plane  by  the  forcing  of 
the  plane  under  the  weight  in  a  direction  parallel  to  the  base  of 
the  plane.  Fig.  31  demonstrates  this. 


FIG.  31. 
Taking  the  ordinary  formula  for  finding  the  force  P  if  friction 

7? 

be  neglected  P  =  W  ™  =  W  tan  a.    It  is  immaterial  whether 

the  weight  be  moved  up  the  plane  or  the  plane  slid  under  the 
weight,  since  P  and  p  are  force  and  reaction.  Taking  friction 
into  account  P=W  tan  (a  +  (f>)  where  </>  is  the  angle  of  repose 
corresponding  to  the  coefficient  of  friction.  Since  it  has  been 
assumed  that  the  coefficient  of  friction  is  .002,  it  follows  that 
the  angle  of  which  this  is  the  tangent  is  1' .  The  following 
table  shows  the  values  of  W  for  different  degrees  of  the  pitch 
angle. 


70 


WORM  GEARS 

TABLE  VIII 


1 

^8 

f  i 

jj 

Jj 

^-^ 

~T~  <^ 

+  0 

^ 

-©. 

a  T* 

Q 

? 

-©- 

«  II 

v-'  II 

<U 

« 

g^ 

(X,  s 

FH 

1 

e 

1 

-f-s  •* 

fel 

•  1 

O 

P- 

II 

a 

£  * 

Ml 

.1 

10° 

10071 

.1784 

178.4 

5609. 

.1763 

15° 

157 

.2701 

270.1 

3700. 

.2679 

20° 

207 

.3662 

366.2 

2732. 

.3639 

25° 

25  7 

.4687 

468.7 

2133. 

.4663 

30° 

307 

.5800 

580.0 

1724. 

.5773 

35° 

357 

.7032 

703.2 

1422. 

.7000 

40° 

407 

.8425 

824.5 

1185. 

.8390 

45° 

457 

1  .  0040 

1004.0 

996. 

1  .  0000 

50° 

507 

1  .  1966 

1196.6 

835. 

1.1917 

55° 

55  7 

1.4343 

1434.3 

698. 

1.4281 

60° 

60  7 

1  .  7402 

1740.2 

575. 

1  .  7320 

The  values  given  in  column  five  would  be  correct  but  for  the 
fact  that  they  take  no  account  of  the  work  done  in  the  form  of 
friction,  and  are  only  indicative  of  the  gain  in  power  due  to  the 
reduction  of  the  worm  gearing,  they  are,  however,  correct  for 
the  angle  of  45  degrees,  as  will  presently  be  shown. 

In  Fig.  32  the  parallelogram  represents  the  developed  sur- 
face of  the  worm,  of  which  A  A  is  the  thread.  The  worm 
wheel  is  to  be  revolved  in  the  direction  LM.  Since  the  teeth 
of  the  wheel  and  the  thread  of  the  worm  are  parallel  to  one 
another  at  the  point  of  contact,  it  follows  that  the  pressure 
between  the  two  is  at  right  angles  to  the  thread  surface.  Let 
aB  represent  this  force,  then  this  may  be  resolved  into  two 
forces  La  and  LB,  of  which  the  latter  is  the  useful  and  the 
former  the  useless  component.  If  the  pitch  of  the  worm  is 
increased  to  CC,  the  pressure  applied  along  the  line  gB  can  be 
resolved  into  gl  and  IB,  of  which  LB  is  the  useful  component; 
Ba  and  Bg  are  by  construction  equal,  and  Bl  is  less  than  BL, 
hence  the  force  which  is  available  for  turning  the  wheel  decreases 
as  the  pitch  is  increased  and  conversely  the  useless  component 
increases  with  the  increasing  angle. 


EFFICIENCY  OF  WORM  GEARING 


71 


BLa  is  a  right  angle  triangle,  of  which  LBa  is  an  angle 
corresponding    to  the  pitch  angle.     Ba  is  the   force  applied. 

D  r 

BL  is  the  useful  force.    The  ratio  of  these  two  -=-  is  the  cosine 

Ba 

of  the  angle  LBa.  Up  to  this  point,  therefore,  it  would  ap- 
pear that,  with  a  constant  worm  torque,  the  force  available 
for  turning  the  wheel  would  vary  inversely  as  the  cosine  of 
the  pitch  angle.  There  is,  however,  another  aspect  which  must 


U 

FIG.  32. 

be  considered,  and  it  is  the  effect  of  the  relative  motion  between 
the  worm  and  wheel.  Thus  with  a  approaching  zero  value 
the  relative  motion  between  the  two  is  very  high  compared  to 
the  useful  motion  of  the  worm  wrheel;  hence  much  work  is 
dissipated  in  useless  friction. 

Various  formula  have  been  proposed  for  determining  the 
actual  mechanical  efficiency  of  worm  gears,  and  so  far  as 
possible,  these  will  be  examined  next. 

We  will  take  first,  the  much  quoted  formula  developed  by 
Professor  Barr  of  Glasgow  University: 


tan  a  (1  —  p  tan  a) 
tan 


(76) 


72  WORM  GEARS 

For  various  angles  of  thread,  a,  the  efficiencies  have  been 
plotted  in  column  II  table  IX. 

Next,  Professor  Unwin  (Elements  of  Machine  Design,  Vol.  1, 
p.  423)  gives  the  following: 

Ji-fi  cot  a  (77) 

1+ju  tan  a 

Column  III  of  the  table  gives  these  values. 

Francis  W.  Davis,  M.E.,  has  proposed  the  following: 


=  1_    / 
\ 


(  (78) 

^cos  a  sin 

Comparing  this  equation  with  (77),  it  will  be  noted  that 
Unwin's  may  be  written 

fi  (cot  a  +  tan  a)  (79) 

1  +  /*  tan  a 

and  this  becomes 

/    _^ 1_  \  (8°) 

r)=l  —  U(-  — T—  .       2          I 

\cos  a  sin  a  +  //  sin  a/ 

very  closely  resembling  (78).  The  values  of  /JL  in  this  equation 
are  given  in  Column  IV. 

It  must  be  observed  here  that  neither  of  the  above  formulae 
takes  into  account  the  pressure  angle,  and  this  undoubtedly 
exercises  a  considerable  influence  upon  the  efficiency  of  the 
gear  inasmuch  as  it  very  largely  governs  the  normal  tooth 
pressure,  quite  irrespective  of  the  torque.  See  equation  43, 
Chapter  VII. 

The  force  of  friction  in  the  gears  is  equal  to  pXp',  and  this 
force  tends  to  resist  the  gliding  of  the  worm  over  the  wheel 
tooth.  It  is  evident  that  the  path  along  which  the  gliding 
occurs  is  a  helical  line  wound  around  a  cylinder  whose  diameter 
equals  the  pitch  diameter  of  the  worm,  and  the  length  of  this 
path  is  I,  the  length  of  thread  per  revolution  measured  at  the 
pitch  line.  See  equation  15,  Chapter  IV.  The  work  lost  in 
friction  equals  in  foot  pounds 


EFFICIENCY  OF  WORM  GEARING 


73 


The  work  put  into  the  worm  is 

Tirdp 

12 
Hence,  the  efficiency  is 


and 
so 


'-^'IVd 

=  7rrt  sec  a 


•  =  1- 


sec  a 


Tnd      /  \       T 

Substituting  the  value  already  found  for  p',  we  get 


T   \2 
(T  cot  a  tan  -^-  1  +  (  rrr~  )  X  sec  a 


This  may  be  written 


u 

=  l  —  -.—         —  \  1  +cos2  a 
sin  a  cos  a  \ 


tan 


(82) 
(83) 

(84) 
(43) 

(85) 
(86) 


In  the  author's  opinion,  this  is  the  most  accurate  formula 
of  the  four  examples,  and  the  values  are  given  in  column  V  of 
the  table  up  to  the  equivalent  lead  angle  of  45  degrees. 

TABLE  IX.— EFFICIENCIES  OF  WORM  GEARING  FOR  VARIOUS 
ANGLES  OF  LEAD 


I 

II 

III 

IV 

V 

a 

g 

. 

g 

n 

15° 

98.5 

99.2 

99.2 

99.0 

20° 

98.68 

99.39 

99.38 

99.3 

25° 

98.8 

99.50 

99.45 

99.4 

30° 

99.0 

99.52 

99.54 

99.4 

35° 

99.5 

99.59 

99.56 

99.53 

40° 

99.5 

99.62 

9.9.59    • 

99.56 

45° 

99.5 

99.63 

99.60 

99.58 

50° 

99.5 

99.63 

99.59 



55° 

99.5 

99.61 

99.55 

60° 

99.5 

99.58 

99.54 

65° 

99.5 

99.50 

99.45 

70° 

99.3 

99.40 

99.38 

74  WORM  GEARS 

It  is  clear,  therefore,  that  for  all  usual  lead  angles  as  em- 
ployed in  automobile  gears,  the  efficiency  is  over  99.5  per  cent. 
and  the  conclusion  is  that  the  whole  question  of  efficiency  is 
one  of  the  simple  equivalent  of  the  work  put  into  the  worm 
minus  that  absorbed  by  friction,  and  may,  for  all  practical  pur- 
poses, be  expressed  thus  for  all  usual  angles 

-**  (87) 


998 
Numerically,  this  is  ^—-  =  99.8  per  cent. 

Some  apology  is  due  to  the  reader  for  introducing  so  simple 
an  expression  as  the  outcome  of  so  cumbersome  an  amount  of 
preliminary  calculation.  The  author  feels,  however,  that 
having  regard  to  the  high  percentage  of  efficiency  this  formula 
gives,  it  would  hardly  be  accepted  by  engineers  unless  the 
previous  proof  were  inserted;  the  demonstration  is  thus 
rendered  complete. 

It  may-  be  pointed  out  that  if  the  table  be  extended  in  both 
directions,  completing  the  series  of  values  of  a  from  0°  to  90°, 
it  will  be  found  that  at  both  limits  the  value  of  ^  is  zero,  which 
is  so  obvious  as  to  need  no  elaboration  further  than  to  observe 
that  if  a  =  0°  or  90°,  there  will  be  no  movement  of  the  worm 
wheel. 

Practical  corroboration  of  these  values  of  r)  has  been  indis- 
putably furnished  by  the  long  series  of  experiments  conducted 
by  Messrs.  Brown  &  Sharpe,  and  published  as  a  memorandum 
by  Professor  Kennerson  in  the  Transactions  of  the  American 
Society  of  Mechanical  Engineers,  1912,  where  average  values  of 
t)  for  the  entire  worm  gear,  including  the  friction  of  all  the 
bearings,  amount  to  over  97  per  cent.,1  and  in  two  instances  in 
the  series  of  experiments,  the  records  of  which  the  author  has 
been  privileged  to  see,  readings  were  obtained  of  99  .  8  per  cent. 

The  efficiencies  given  in  column  V  of  Table  IX  may,  there- 
fore, be  accepted  as  representing  what  may  with  care  be  ob- 
tained in  practical  working,  provided  the  workmanship  is 

xAlso  corroborated  by  Nat.  Phys.  Lab.,  London  in  tests  on  Daimler 
worm-gear,  Nov.,  1912,  (seeaute). 


EFFICIENCY  OF  WORM  GEARING  75 

good,  and  the  mounting  and  lubrication  of  the  gears  properly 
carried  out. 

The  late  Mr.  Briggs,  of  Philadelphia,  says  in  regard  to 
friction,  "It  is  established  that  for  ordinary  ratio  of  wheel  to 
worm,  say  not  to  exceed  60  or  80  to  1,  well-fitted  worm  gear  will 
transmit  motion  backward  through  the  worm,  exhibiting  a  lower 
coefficient  of  friction  than  is  found  in  any  other  description  of 
running  machinery." 

The  selection  of  .002  as  the  value  of  the  coefficient  of  friction, 
earlier  in  this  chapter  is  thus  justified. 


CHAPTER  XI 
GENERAL  POINTS  OF  DESIGN  OF  MOUNTING. 

The  design  of  the  worm  and  wheel  have  now  been  fully  con- 
sidered, and  the  stresses  set  up  in  the  various  bearings  in- 
vestigated to  the  point  where  the  designer  is  in  possession  of  all 
the  information  to  enable  him  to  design  the  casing  in  which  the 
gears  will  work.  For  a  worm  gear  which  is  to  work  stationary 
machinery  the  provision  of  a  suitable  casing  is  a  comparatively 
simple  matter  the  only  conditions  being  rigidity  and  the  provi- 
sion of  an  oil-tight  casing  to  carry  the  lubricant.  The  quality 
of  rigidity  must  be  insisted  upon  to  ensure  the  maintenance  of 
the  correct  relative  positions  of  the  worm  and  wheel  under  the 
heavy  stresses  to  which  they  are  subjected.  Weight  in  such 
cases  is  of  secondary  importance  and  metal  need  not  be  spared 
to  ensure  the  desired  end.  The  casing  moreover  will  in  such  a 
case  be  bolted  securely  to  a  solid  foundation  which  greatly 
facilitates  matters. 

Very  different  are  the  conditions  in  an  automobile  axle  where 
weight  must  be  saved  at  any  cost,  and  strength  and  lightness 
have  both  to  be  studied. 

In  the  days  of  the  first  Lanchester  Car,  8  H.P.  was  all  that 
had  to  be  provided  for,  and  the  weight  of  the  car  was  but  a  few 
hundred  pounds.  Recent  worm  gears  the  author  designed  are 
capable  of  transmitting  90  H.P.  in  a  fast  pleasure  car  which 
weighs  over  three  tons  with  a  load  of  seven  passengers  and  their 
luggage;  at  the  time  of  writing  it  is  believed  that  these  gears  are 
considerably  more  powerful  than  any  others  working  in  auto- 
mobiles, although  their  size  is  by  no  means  excessive.  It  may 
well  be  imagined  that  the  provision  of  a  rigid  case  for  these  is 
something  of  a  problem;  the  result  has,  however,  proved 
entirely  satisfactory. 

In  the  first  place  there  is  the  question  of  whether  the  worm 
shall  be  placed  above  or  below  the  wheel.  This  subject  has 

76 


GENERAL  POINTS  OF  DESIGN  OF  MOUNTING   77 

been  discussed  by  different  makers  as  though  it  were  a  matter 
of  great  moment;  in  reality  it  is  of  no  consequence  whatever, 
and  is  purely  to  be  governed  by  the  expediency  of  other  factors 
in  the  general  design.  Much  more  important  is  the  design  of 
the  axle  casing.  Based  no  doubt  upon  bevel  gear  practice,  the 
casing  was  originally  made  in  two  halves,  each  containing  half 
the  bearings  required  to  support  the  differential,  which  were 
bolted  together  around  the  center,  the  portion  of  the  case  con- 
taining the  worm  was  also  in  halves,  being  in  fact  an  extension 
of  the  main  casting.  Perfectly  satisfactory  axles  have  been 
built  on  this  plan  but  the  difficulties  of  assembling  are  very 
great  as  neither  the  worm  nor  the  wheel  can  be  seen  after  they 
are  mounted  in  position.  A  more  modern  type  is  shown  in 
Fig.  33  which  is  an  illustration  of  a  rear  axle  of  a  five-ton  com- 
mercial vehicle,  here  reproduced  by  permission  of  the  Fierce- 
Arrow  Motor  Car  Company.  It  will  be  seen  that  the  axle 
proper  is  a  pan-shaped  casting,  with  elongations  which  carry  the 
sjteel  tubular  extensions  on  which  the  road  wheels  are  mounted. 
At  the  outset  therefore  a  very  rigid  structure  is  provided  for 
preserving  the  true  alignment  of  the  driving  shafts.  The  worm 
and  the  wheel  are  mounted  in  a  single  casting,  provided  with 
strongly  ribbed  brackets  for  carrying  the  bearings  of  the  worm 
wheel,  the  whole  forming  the  lid  of  the  axle  case  proper.  By 
this  arrangement  the  assembling  of  the  gears  and  their  exact 
adjustment  can  be  carried  out  on  the  bench  with  the  worm  and 
wheel  in  full  view,  giving  every  facility  to  the  erector  for  correct 
location  in  their  relative  positions;  such  a  method  also  enables  an 
accurate  machining  operation  to  be  carried  out  upon  the  main 
bearings  of  the  gear,  and,  when  examination  is  necessary,  it 
can  be  made  without  disturbing  any  adjustments,  the  entire 
system  of  worm  wheel,  and  differential  with  all  their  bearings 
being  lifted  out  in  one  piece  for  the  purpose. 

Much  latitude  is  permissible  to  the  designer,  who  has  a  wide 
field  of  arrangements  to  select  from;  it  must  always  be 
remembered  that  the  stresses  in  the  parts  supporting  the  bear- 
ings are  high,  but  with  the  use  of  the  diagram,  Fig.  19,  they 
may  be  determined  at  a  glance,  and  a  selection  of  suitable 
ball  bearings  can  be  made. 


78 


WORM  GEARS 


GENERAL  POINTS  OF  DESIGN  OF  MOUNTING    79 

It  may  be  observed  that  in  the  case  of  a  straight  worm,  that 
is,  one  having  a  cylindrical  pitch  line,  a  double  thrust  bearing 
should  be  provided  at  one  end  (whichever  is  most  convenient) 
the  other  end  can  then  be  left  free  to  expand  or  contract  with 
the  difference  of  temperature  which  occurs  when  the  worm  is 
running.  In  the  case  of  the  hour-glass  pattern  worm,  it  is 
difficult  to  say  what  happens  when  it  has  to  expand,  presumably 
the  casing  expands  too,  and  in  that  case  a  single  thrust  bearing 
at  either  end  is  the  better  arrangement. 

Some  designers  provide  a  very  heavy  thrust  bearing  to  take 
the  forward  drive  and  a  relatively  light  one  for  the  reverse — 
it  is  hard  to  see  any  justification  for  such  an  arrangement,  since 
the  reverse  gear  is  almost  invariably  lower  than  the  first  speed 
forward,  and  the  torque  at  the  worm  pitch  line,  and  all  the 
resultant  pressures  are  in  consequence  heavier. 

Provision  must  be  made  for  oil  to  reach  the  thrust  bearing 
at  the  rear  of  the  worm  when  the  latter  is  mounted  above  the 
wheel.  The  simplest  way,  and  one  which  can  be  confidently 
recommended,  is  to  bore  the  worm  spindle  through  the  center 
and  drill  small  holes,  say  3/16-in.  diameter,  radially  into  the 
hollow  center,  these  holes  being  of  course  drilled  between  the 
threads  of  the  worm  before  it  is  hardened.  In  the  case  of  a 
large  worm  for  high  powered  pleasure  cars,  this  method  has  the 
added  advantage  of  considerably  lightening  the  worm,  and  the 
hollow  in  the  spindle  carries  the  oil  back  to  the  thrust  bearing  in 
a  steady  stream. 

Provided  precautions  are  taken  to  prevent  the  oil  from 
leaking  out  through  the  arms  of  the  axle,  it  will  be  found  that 
the  same  oil  may  be  used  almost  indefinitely — at  any  rate 
from  5000  to  7000  miles,  and  it  may  be  observed  that  the  oil 
level  should  be  somewhat  below  the  center  of  the  casing.  A 
small  vent  pipe  may  be  provided  if  thought  necessary,  to  allow 
for  expansion  of  the  air  when  the  gears  get  warm,  this  will 
prevent  the  oil  being  forced  out  round  the  axles  by  any  pres- 
sure so  formed;  such  a  pipe  must,  however,  be  closed  by  a  cap 
drilled  with  a  very  small  hole,  say  1/16-in.  diameter  and  the 
pipe  should  be  loosely  plugged  with  cotton  to  prevent  the 
entrance  of  dust. 


80  WORM  GEARS 

The  author  has  intentionally  refrained  from  elaborating  to 
any  extent  upon  the  details  of  the  casing,  which  are  best  left 
to  the  judgment  of  the  designer;  sufficient  attention  has, 
however,  been  drawn  to  the  more  important  features  to  enable 
a  satisfactory  design  to  be  produced  if  the  principles  set  out  in 
the  earlier  chapters  of  this  book  are  closely  followed.  In  con- 
clusion, let  it  be  pointed  out  that  no  workmanship  can  be  too 
accurate  in  the  manufacture  of  worm  gearing,  and  unless 
facilities  exist  for  this,  a  satisfactory  result  cannot  be  looked  for. 
If,  on  the  other  hand,  proper  precautions  are  taken,  a  worm  gear, 
correctly  designed  and  mounted,  is,  as  Mr.  Briggs  has  observed, 
probably  the  most  efficient  piece  of  machinery  known;  and  its 
efficiency  is  only  equalled  by  its  durability  and  silence  of 
operation. 


CHAPTER  XII 

RECAPITULATION  OF  FORMULAE  USED 

Wheel  circumference 


Gear  ratio  G  = 


worm  lead 

-  (5) 


Measurement  of 

(a)  Axial  pitch  =P 

(b)  Normal  pitch  P'  =  P  cos  a  \  (2) 

(c)  Circumferential  pitch     P"  =  P  cot  a  \ 
Circumference  of  worm  =  nd-=nP  cot  a  (3) 


Number  of  threads  on  worm  =  ?i=  -  (4) 

P  cot  a 

Lead=L=^  (6) 

L  =  Pn  (14) 

Lead  angle  =  a 


tan  a 

L 

(7) 

tan  a 

=  £ 

(8) 

tan  a 

Pn 
~xd 

(17) 

Length  of 


g  -    2/g^  sin  -  (9) 

81 


82  WORM  GEARS 

Number  of  teeth  in  contact  =  n' 


(11) 


sn 


P 

Centers  of  worm  and  wheel  =  c 


Length  of  thread  per  revolution  of  worm 

=  l  =  7id  sec  a  (15) 

Rubbing  velocity  of  worm  =  7; 

ltd  sec  ap  n  a. 

v  =  —  =    —  ft.  per  second 


Ratio  between  axial  pressure  angle  0  and  normal  pressure 
angle  ¥ 

¥ 

Q      tan-9- 

tan~~-  (19) 

2      cos  a 


tan—  =  tan  -^  cos  a  (20) 

_  -- 

Normal  tooth  pressure  =  pf 


T 

p'=~  (26) 

cos 


sin(90°-/) 


27    '  Vsin  a 


RECAPITULATION  OF  FORMULA  USED          83 

Resultant  normal  pressure  angle  =  z 

¥ 

z  =  cos  a  tan-  (46) 

Tangential  tooth  pressure  (due  to  tractive  resistance) 

wu.R'  /OQX 

=  p  =  — W—  (*°) 

Length  of  wheel  tooth =^ 

*-m  (29) 

Safe  load  on  wheel  tooth  =  x 

x  =  2  2  ISP  (34) 

Worm  end  thrust  =  p 

p=T  cot  a  (49) 

Separating  force  between  worm  and  wheel  =/ 

W 

f=  T   cot  a  tan  -  (53) 

^ 

Radial  thrust  on  worm  bearings  =  S 


T*  (56) 

/ 

Subtended  angle  of  worm  =  /? 

i*j 

(59) 


2.2  IPn' 

360 


~  TI  (d  +.6366  P) 
Tooth  pressure  velocity  constant  =  p'v 

p'v  =  17000  (67) 

Heat  generated  in  gears  =  H 

'iv  60  ,  ~ 

(68) 


84  WORM  GEARS 

Area  of  axle  surface  required  to  dissipate  heat  =  a 

1383 
a-—,— 
p'j® 

Coefficient  of  friction  =  /* 

^  =  .  002  =  tan  a  =  tan  1'  (75) 

Efficiency  (accurate)  =  t] 


Efficiency  (approximate) 

,-  ^  (87) 


INDEX 


Addendum,  10,  16 
Annealing  worms,  7 
Area  of  physical  contact,  50,  52 
Axial  pitch  of  worm,  10 

Bach  and  Roser,  60,  64 
/?  value  of,  48 

/?  variations  of,  author's  experi- 
ments, 52 

Bronze  alloy  for  worm  wheel,  5 
Brown  &  Sharpe,  efficiency  experi- 
ments, 74 
Bruce,  Robert  A.,  deductions,  50 

Case  hardening,  worms,  6 

Casing,  design  of,  76 

Centers  of  axes,  19 

Circular  pitch  of  wheel,  definition  of, 

10 
Circumferential  pitch,  definition  of, 

10 


Heat  generated,  58 
Hindley  worm  gear,  7,  8,  29 
Hobbing,  6 

Included  angle,  calculations  of,  25 

of  thread,  definition  of,  11 
Interference  of  wheel  teeth,  22 

Kennerson,  Professor,  on  Brown  & 
Sharpe  experiments,  74 

Lanchester,  early  worm  gear,  2,  8 

gear,  30 

Lead  angle,  calculations  for,  17,  19 
definition  of,  10 

definition  of,  9 

of  worm,  19 
Length  of  teeth,  calculation  of,  27 

of  thread,  19 

of  worm,  definition  of,  10 

determination  of,  17 
Lost  work  of  worm,  21 
Lubricating  oil,  57 


Daimler  axle,  test  of,  33,  74 
Dedendum,  10,  16 
Dennis,  early  worm  gear,  2 

Manufacturing  methods,  6,  7 
Efficiency,  Daimler  worm  axle,  74,      Materials  for  worm  gears,  5,  6 

foot  note. 

formula  for,  72,  73,  74 
'    of  worm  gearing,  63,  et  seq. 


Mechanical  efficiency,  72,  73,  74 
Normal  pitch,  definition  of,  10 
Oil,  film,  56 


lubricating,  57 
Fierce-Arrow  Motor  Car  Co.  worm 


Forms  of  teeth,  29 
Friction,  Beauchamp  Tower  experi- 
ments, 63 

coefficient  of,  63 

in  axle,  author's  experiment,  67  axles,  2,  77 

Pitch,  calculation  of,  17 
Gear  ratio,  calculations  for,  17 

definition  of,  8 
Gliding  angle,  26 
Globoid  gear,  3 


Hardness  of  materials  used,  5,  6 


in  relation  to  horsepower,  14 
line  of  wheel,  definition  of,  9 
of  worm,,  definition  of,  10 
selection  of,  13 
Pressure  angle,  22 
definition  of,  11 


85 


86 


INDEX 


Proportions  of  gears. 

Rack,  8 

Radiation  of  heat  from  axle,  59 
Reversibility,  8,  23 
Rubbing  velocity,  19 

Screw,  definition  of,  1 
Strength  of  bronze  alloy,  5 

of  steel  for  worm,  6 

of  teeth,  34 

Stresses  in  gears,  38-46 
Subtended  angle,  definition  of,  11 
Symbols,  11,  12 

Temperature,  coefficient,  58 
maximum  permissible,  58 
of  gears,  51 

Teeth  in  contact,  number  of,  17 


Thread,  form  of,  22 

proportions  of,  26 
Threads,  number  of,  17 
Tooth  pressure,  effect  of  varying,  62 

pressures,  calculations  of,  28 
Tower,  Beauchamp,  friction  experi- 
ments, 63,  68 
Tractive  resistance,  34 

Velocity,  rubbing,  effect  of  varying, 

62 
Viscosity  of  lubricant,  52 

Width  of  worm  wheel,  47 
Worm  gearing,  definition  of,  8 

location  of,  77 

material  for,  6 

wheel,  material  for,  5 
size,  determination,  16 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


MAR   111935 
LIBRARY  USE 

Mt\T*~1 

.OK  2  1935  ^UL  10  1957 

tter^n  ID 

f*     1Q39 

MM\I          \J          *^|J 

APR  30  1942 

:>CT     3    1943 

=.      tlApr'SO^i 

,  v  J  s        u 

-" 

LD  21-100m-8,'34 

